-8pt

# Global and local observability of hypergraphs

Chencheng Zhang1, Hao Yang1, Shaoxuan Cui2, Bin Jiang1,
and Ming Cao3 The preprint is submitted to IEEE CDC 2024 on 20 March, 2024.
This work was supported by the National Natural Science Foundation of China
under Grants 62073165 and 62233009.1 C. Zhang, H. Yang and B. Jiang are with
the College of Automation Engineering, Nanjing University of Aeronautics and
Astronautics, Nanjing, 211106, China. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>(Corresponding author: Hao Yang) 2
S. Cui is with the Bernoulli Institute for Mathematics, Computer Science and
Artificial Intelligence, University of Groningen, Groningen, 9747 AG
Netherlands. E-mail<EMAIL_ADDRESS>M. Cao is with the Engineering and
Technology Institute Groningen, University of Groningen, Groningen, 9747 AG
Netherlands. E-mail<EMAIL_ADDRESS>
###### Abstract

This paper studies observability for non-uniform hypergraphs with inputs and
outputs. To capture higher-order interactions, we define a canonical non-
homogeneous dynamical system with nonlinear outputs on hypergraphs. We then
construct algebraic necessary and sufficient conditions based on polynomial
ideals and varieties for global observability at an initial state of
hypergraphs. An example is given to illustrate the proposed criteria for
observability. Further, necessary and sufficient conditions for local
observability are derived based on rank conditions of observability matrices,
which provide a framework to study local observability for non-uniform
hypergraphs. Finally, the similarity of observability for hypergraphs is
proposed using similarity of tensors, which reveals the relation of
observability between two hypergraphs and helps to check the observability
intuitively.

###### Index Terms:

Observability, hypergraphs, higher-order interactions, algebraic approaches.

##  I Introduction

Over the past two decades, the emergence and evolution of network science have
helped unveil the underlying mechanisms of network systems with
interdisciplinary applications including systems engineering, systems biology,
genetics, and neuroscience [1]. Most research subjects focus on networks
inherently depicted by only interactions between pairs of nodes [2], whereas,
in reality, the interactions within many systems go beyond dyadic
relationships. It has been shown that functional brain networks [3], protein
interaction networks [4], and co-authorship graphs of publications [5] cannot
be simply factorized by pairwise interactions. Instead, they involve group-
wise, or so-called higher-order, interactions among the nodes in the network.
To bridge this gap, hypergraphs, where hyperedges link multiple nodes, are a
potentially more powerful modeling tool to represent such kind of interactions
[6]. For example, hypergraphs can be utilized to model transportation systems
where routes involve multiple locations or intersections, allowing a more
realistic depiction of traffic flows [7]. Therefore, investigating the
performance of hypergraphs has emerged at the forefront of research in network
dynamics.

As one of the most important properties of a dynamical system, observability
is a measure that tells whether the trajectory temporal evolution of the
internal states of the system can be reconstructed from the knowledge of
inputs and outputs [8]. It can further indicate how to place sensors, as few
as possible, to determine states [9]. As for network systems, it can help
position sensors on nodes to observe the states of all nodes. Global and local
observability of an initial state are defined according to whether the initial
state is distinguishable from all the possible initial states or initial
states in one of its neighborhood. Most existing theoretical analysis for
general nonlinear systems rely on algebra and differential geometry. Usually,
the resulting approaches propose sufficient conditions for observability by
computing the dimension of the subspace spanned by the gradients of the Lie
derivatives of the measurements [10][11], which leads to the infinite number
of Lie derivatives, especially for the global observability criterion for
general nonlinear systems [12]. However, for polynomial systems, it has been
proved that global observability can be characterized by a finite set of
equations based on commutative algebra [13]. This can be further extended to
analyze a class of nonlinear systems that can be transformed into polynomial
expressions.

To the best of our knowledge, there are very little work focusing on the
observability of hypergraphs. Ref. [14] investigates local observability of
uniform hypergraphs. However, it does not consider global observability and
non-uniform hypergraphs. In this paper, we define a tensor-based dynamics on
non-uniform hypergraphs with inputs and outputs and analyze both global and
local observability criteria. The main contributions of our work are as
follows:

  * •

A generalized model of network dynamics with inputs and outputs on non-uniform
hypergraphs is proposed based on the tensor-vector multiplication property.
Specifically, we investigate non-uniform hyperedges consisting of nodes both
in system dynamics and outputs, which have not been considered in existing
research.

  * •

We represent a general non-homogeneous polynomial system in a tensor form,
which can be further studied from a perspective of hypergraphs. Sufficient and
necessary conditions for global observability at an initial state are derived
based on properties of polynomial ideals and varieties. These conditions
construct an effective tool to test the global observability of hypergraphs
accurately within finite steps.

  * •

To obtain observability criteria of hypergraphs based on the system
characteristics, Kalman-rank-like conditions for local observability are
proposed, which reveal connections between system parameters and
observability.

  * •

The similarity of observability for higher-order systems is proposed based on
the similarity of tensors, which reveals the relationship between two similar
higher-order systems from the perspective of observability, this helps to
intuitively figure out the observability of more hypergraphs after checking
just one.

The rest of the paper is organized as follows: Section II gives preliminaries.
Section III constructs general hypergraph dynamics and analyzes global and
local observability criteria for hypergraphs. Section IV gives conclusions.

##  II Preliminaries

###  II-A Tensors

A tensor is a multidimensional array. The order of a tensor is the number of
its dimensions, and each dimension is called a mode. A k-th order tensor
usually is denoted by $\mathbf{T}\in\mathbb{R}^{n_{1}\times
n_{2}\times\cdot\cdot\cdot\times n_{k}}$ and it is called cubical if all modes
have same size, i.e., $n_{1}=n_{2}=\cdot\cdot\cdot=n_{k}=n$. A cubical tensor
T is called super-symmetric if $\mathbf{T}_{j_{1}j_{2}\cdot\cdot\cdot j_{k}}$
is invariant under any permutation of the indices.

Definition 1 [15]:  The tensor vector multiplication
$\mathbf{T}\times_{p}\mathbf{v}$ along mode $p$ for a vector
$\mathbf{v}\in\mathbb{R}^{n_{k}}$ is defined as

$\displaystyle(\mathbf{T}\times_{p}\mathbf{v})_{j_{1}j_{2}\cdot\cdot\cdot
j_{p-1}j_{p+1}\cdot\cdot\cdot
j_{k}}=\sum_{j_{p}=1}^{n_{p}}\mathbf{T}_{j_{1}j_{2}\cdot\cdot\cdot j_{p}\cdots
j_{k}}\mathbf{v}_{j_{p}},$ (1)

which can be extended to

$\displaystyle\mathbf{T}\times_{1}\mathbf{v}_{1}\times_{2}\mathbf{v}_{2}\cdot\cdot\cdot\times_{k}\mathbf{v}_{k}=\mathbf{T}\mathbf{v}_{1}\mathbf{v}_{2}\cdot\cdot\cdot\mathbf{v}_{k}\in\mathbb{R}$
(2)

where $\mathbf{v}_{p}\in\mathbb{R}^{n_{p}}$ with $p\in\\{1,...,k\\}$. For
simplicity, we define
$\mathbf{T}\mathbf{v}_{1}\mathbf{v}_{2}\cdot\cdot\cdot\mathbf{v}_{k}:=\mathbf{T}\mathbf{v}^{k}$
if $\mathbf{v}_{1}=\cdots=\mathbf{v}_{k}=\mathbf{v}$.

###  II-B Hypergraphs

Definition 2 [16]: Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a hypergraph
with $n$ nodes and $h$ hyperedges. Define $e_{ki},{ki}\in\\{1,\cdots,h\\}$ as
the $i$-th hyperedge with the cardinality $k$ where $i=1,2,\cdots$. Define the
tensor $\mathbf{A}_{k}$ as follows

$\displaystyle\mathbf{A}_{k}(a_{1}...a_{k}):=\left\\{\begin{array}[]{cc}\frac{1}{k-1},\
\ (a_{1}...a_{k})\in\cup_{i=1,2,\cdots}e_{ki},\\\ 0,\ \ \ \
(a_{1}...a_{k})\notin\cup_{i=1,2,\cdots}e_{ki}.\end{array}\right.$ (5)

###  II-C Observability of nonlinear systems

Consider a nonlinear system in the following form

$\displaystyle\dot{x}$ $\displaystyle=F(x,u),x(0)=x_{0},$ $\displaystyle y$
$\displaystyle=H(x,u).$ (6)

where $x\in\mathbb{R}^{n}$, $u\in\mathbb{R}^{m}$, and $y\in\mathbb{R}^{q}$
denote state, input, output vectors.
$F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and
$H:\mathbb{R}^{n}\rightarrow\mathbb{R}^{q}$ are real analytic functions.

Let us recall some definitions of observability[17][18].

Definition 3 : A pair of initial states
$(\xi,\eta)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ of the system (6) is
distinguishable if there exists an admissible piecewise constant input $u$ and
an instant $t$ such that $y(t;\xi,u)\neq y(t;\eta,u)$.

Definition 4 : The system (6) is globally observable at an initial state
$\xi\in\mathbb{R}^{n}$ if and only of for any
$\eta\in\mathbb{R}^{n}\setminus\\{\xi\\}$, the pair of $(\xi,\eta)$ is
distinguishable.

Definition 5: The system (6) is globally observable if and only if every pair
of distinct initial states $(\xi,\eta)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$
is distinguishable.

##  III Observability Criterions of hypergraphs

###  III-A General hypergraph dynamics

Consider a hypergraph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $n$ nodes
and $h$ hyperedges. Let $c$ be the maximum cardinality of the hyperedges.
Define a general input-affine higher-order interacted system under hypergraph
$\mathcal{G}$, whose dynamics are as follows

$\displaystyle\begin{array}[]{cc}\dot{{x}}=&\sum_{k=2}^{c}\mathbf{A}_{k}{x}^{k-1}+\sum_{j=1}^{m}\sum_{k=2}^{c}\mathbf{B}_{k,j}{x}^{k-1}u_{j}\\\
y_{i}=&\\!\\!\\!\\!\sum_{k=1}^{c}\mathbf{C}_{i,k}{x}^{k}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathbf{D}_{i,k,l}{x}^{k}u_{l},\end{array}$
(9)

where $i=1,\cdots,q$, ${x}:=(x_{1},x_{2},...,x_{n})^{T}$, and
${x}(0)={x}_{o}$. $u_{j}$ and $y_{i}$ are the $j$-th input and the $i$-th
output where $j=1,\cdots,m$ and $i=1,\cdots,q$.
$\mathbf{A}_{k},\mathbf{B}_{k},\textbf{C}_{i,k},\mathbf{D}_{i,k,l}\in\mathbb{R}^{n\times
n\times\cdots\times n}$ denote $k$-th order system, input, output and direct
transmission tensors, respectively. Define $e_{\mathbf{A}kr}$,
$e_{\mathbf{B}kjr}$, $e_{\mathbf{C}ikr}$ and $e_{\mathbf{D}iklr}$ as the
$i$-th hyperedge with $k$ cardinality where $r=1,2,\cdots$ in coupled dynamics
of subsystems corresponding to the system ${x}$, input $u_{j}$, output $y_{i}$
and direct transmission $u_{l}$ for output $y_{i}$, respectively.
$\mathbf{A}_{k},\mathbf{B}_{k,j},\mathbf{C}_{i,k}$ and $\mathbf{D}_{i,k,l}$
are defined as follows

$\displaystyle\mathbf{A}_{k}(a_{1}\cdots
a_{k}):=\left\\{\begin{array}[]{cc}&\frac{1}{(k-1)!},\ \
(a_{1},\cdots,a_{k})\in\cup_{i=1,2,\cdots,e_{\mathbf{A}kr}},\\\ &0,\ \
\mathbf{otherwise}\end{array}\right.$ (12)

$\displaystyle\mathbf{B}_{k,j}(b_{1}\cdots
b_{k}):=\left\\{\begin{array}[]{cc}&\frac{1}{(k-1)!},\ \
(b_{1},\cdots,b_{k})\in\cup_{i=1,2,\cdots,e_{\mathbf{B}kjr}},\\\ &0,\ \
\mathbf{otherwise}\end{array}\right.$ (15)

$\displaystyle\mathbf{C}_{i,k}(c_{1}\cdots
c_{k}):=\left\\{\begin{array}[]{cc}&\frac{1}{k!},\ \
(c_{1},\cdots,c_{k})\in\cup_{i=1,2,\cdots,e_{\mathbf{C}ikr}},\\\ &0,\
\mathbf{otherwise}.\end{array}\right.$ (18)

$\displaystyle\mathbf{D}_{i,k,l}(d_{1}\cdots
d_{k}):=\left\\{\begin{array}[]{cc}&\frac{1}{k!},\ \
(d_{1},\cdots,d_{k})\in\cup_{i=1,2,\cdots,e_{\mathbf{D}iklr}},\\\ &0,\
\mathbf{otherwise}.\end{array}\right.$ (21)

Note that system (9) can be represented in its Kronecker form as

$\displaystyle\begin{array}[]{cc}\dot{{x}}=&\sum_{k=1}^{c}\mathcal{A}_{k}{x}^{[k-1]}+\sum_{j=1}^{m}\sum_{k=1}^{c}\mathcal{B}_{k,j}{x}^{[k-1]}u_{j},\\\
y_{i}=&\\!\\!\\!\\!\\!\\!\\!\\!\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}{x}^{[k]}u_{l},\end{array}$
(24)

where $i=1,\cdots,q$ and ${x}(0)={x}_{o}$.
$\mathcal{A}_{k}=\mathbf{A}_{k}\in\mathbb{R}^{n\times n^{k-1}}$,
$\mathcal{B}_{k,j}=\mathbf{B}_{k,j}\in\mathbb{R}^{n\times n^{k-1}}$,
$\mathcal{C}_{i,k}=\mathbf{C}_{i,k}\in\mathbb{R}^{1\times n^{k}}$, and
$\mathcal{D}_{i,k,l}=\mathbf{D}_{i,k,l}\in\mathbb{R}^{1\times n^{k}}$ are
unfolding of $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$.
Moreover, for simplicity of notation, the system (24) can be denoted as the
following integrated form

$\displaystyle\begin{array}[]{cc}\dot{{x}}=&f({x},u),\ \ {x}(0)={x}_{o}\\\
y_{i}=&h_{i}({x},u),\ \ i=1,2,\cdots,q\end{array}$ (27)

where

$\displaystyle\begin{array}[]{cc}f({x},u):=g_{0}({x})+\sum_{j=1}^{m}g_{j}({x})u_{j}\\\
h_{i}:=p_{i,0}({x})+\sum_{l=1}^{w_{i}}p_{i,l}({x})u_{l}\end{array}$ (30)

with $g_{0}({x}):=\sum_{k=1}^{c}\mathcal{A}_{k}{x}^{[k-1]}$,
$g_{j}({x}):=\sum_{k=1}^{c}\mathcal{B}_{k,j}{x}^{[k-1]}$ for $j=1,\cdots,m$
and $p_{i,0}({x}):=\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]}$ and
$p_{i,l}({x}):=\sum_{k=1}^{c}\mathcal{D}_{i,k,l}{x}^{[k]}$ for
$l=1,\cdots,w_{i}$.

Remark 1:  The non-uniform hypergraph (9) is modeled by dividing it into
several uniform hypergraphs described by tensors with different dimensions.
For every $k\in\\{1,\cdots,c\\}$, $\mathbf{A}_{k}{x}^{k-1}$ can be viewed as
all the hyperedges with $k$ nodes in the hypergraph. $\Box$

###  III-B Global observability of hypergraphs

Lemma 2 : A pair of initial states $(\xi,\eta)$ is indistinguishable if and
only if, for all $\tau_{1},\tau_{2},\cdots,\tau_{r}\in\\{g_{0},g_{1},\cdots
g_{m}\\},r=0,1,\cdots$ and
$\upsilon_{c}\in\\{p_{i,0},p_{i,1},\cdots,p_{i,w_{i}}\\},i=1,\cdots,q$,
$c=0,\cdots,\sum_{i=1}^{q}(w_{i}+1)-1$, it holds that

$\displaystyle L_{\tau_{1}}L_{\tau_{2}}\cdots
L_{\tau_{r}}\upsilon_{c}(\xi)=L_{\tau_{1}}L_{\tau_{2}}\cdots
L_{\tau_{r}}\upsilon_{c}(\eta)$ (31)

where the Lie derivative $L_{\tau}v:=\frac{\partial v}{\partial x}\tau$.

Proof : For system (9), consider the following piecewise constant input

$\displaystyle u_{k}=\left\\{\begin{array}[]{ccc}u_{k1}&t\in[0,t_{1})\\\
u_{k2}&t\in[t_{1},t_{1}+t_{2})\\\ \vdots&\end{array}\right.$ (35)

where $t_{1},t_{2},\cdots$ are assumed to be sufficiently small. Since
$h_{1},h_{2},\cdots,h_{q}$ are real analytic functions of $t$, the output
$y_{i}$ at time $t\in\\{t_{1},t_{1}+t_{2},\cdots\\}$ can be computed in the
following Taylor series form

$\displaystyle y_{i}(t_{1})$
$\displaystyle=h_{i}(x_{0},u_{k1})+L_{f(x,u_{k1})}h_{i}(x,u_{c1})|_{x_{0}}t_{1}+\cdots,$
$\displaystyle y_{i}(t_{1}+t_{2})$
$\displaystyle=h_{i}(x_{0},u_{k_{1}})+L_{f(x,u_{k1})}h_{j}(x,u_{k1})|x_{0}t_{1}$
$\displaystyle\hskip
14.22636pt+L_{f(x,u_{k2})}h_{j}(x,u_{k2})|_{x_{0}}t_{2}+\cdots,$
$\displaystyle\hskip 11.38092pt\vdots$

Since system (24) can be computed as a polynomial system, the functions $f,h$
are real analytic. Therefore, a pair of initial states $(\xi,\eta)$ is
indistinguishable if and only of for all $i=1,\cdots,q$, it holds that

$\displaystyle\begin{array}[]{ccc}&h_{i}(x,u_{k1})|_{\xi}=h_{i}(x,u_{k1})|_{\eta},\\\
&L_{f(x,u_{k1})}h_{i}(x,u_{c1})|_{\xi}=L_{f(x,u_{k1})}h_{i}(x,u_{c1})|_{\eta},\\\
&\vdots\end{array}$ (39)

Since for any possible $u$, the above equations (39) should hold. It follows
from (30) that equations (39) lead to

$\displaystyle\begin{array}[]{ccc}&p_{i,k}(\xi)=p_{i,k}(\eta),\\\
&L_{g_{r}}p_{i,k}(\xi)=L_{g_{r}}p_{i,k}(\eta),\\\ &\vdots\end{array}$ (43)

with $r=0,1,\cdots,m$ and $k=0,1,\cdots,w_{i}$, which can be incorporated as
condition (31). $\Box$

According to Lemma 2, one can conclude that the set of initial states that
satisfy condition (31) are not observable. However, it is difficult to
determine how many Lie derivatives are sufficient to test the observability.
To solve this problem, relying on Hilbert’s basis theorem, the following
observability criteria can be obtained using a finite set of equations by
using properties of polynomial systems [13].

From condition (31), if the initial state pair $(\xi,\eta)$ is
indistinguishable, it holds that

$\displaystyle v_{c}(\xi)$ $\displaystyle=v_{c}(\eta),$ $\displaystyle
L_{g_{j}}v_{c}(\xi)$ $\displaystyle=L_{g_{j}}v_{c}(\eta),$ $\displaystyle
L_{g_{k}}L_{g_{j}}v_{c}(\xi)$ $\displaystyle=L_{g_{k}}L_{g_{j}}v_{c}(\eta),$
$\displaystyle\hskip 5.69054pt\vdots$

where $c=1,\cdots,\sum_{i=1}^{q}(w_{i}+1)$, $j\in 1,\cdots,m$, and $k\in
1,\cdots,m$. For $i=1,\cdots,q$ and $r=0,1,\cdots$, define the ideals
$J_{r,i}$ as

$\displaystyle J_{0,i}$
$\displaystyle=<v_{i_{1}}(\xi)-v_{i_{i}}(\eta),\cdots,v_{i_{s_{i}}}(\xi)-v_{i_{s_{i}}}(\eta)>$
$\displaystyle J_{1,i}$
$\displaystyle=J_{0,i}+<L_{g_{1}}v_{i_{1}}(\xi)-L_{g_{1}}v_{i_{1}}(\eta),$
$\displaystyle\hskip
85.35826pt\cdots,L_{g_{m}}v_{i_{s_{i}}}(\xi)-L_{g_{m}}v_{i_{s_{i}}}(\eta)>$
$\displaystyle J_{2,i}$
$\displaystyle=J_{1,i}+<L_{g_{1}}L_{g_{1}}v_{i_{i}}(\xi)-L_{g_{1}}L_{g_{1}}v_{i_{1}}(\eta),$
$\displaystyle\hskip
56.9055pt\cdots,L_{g_{m}}L_{g_{m}}v_{i_{s_{i}}}(\xi)-L_{g_{m}}L_{g_{m}}v_{i_{s_{i}}}(\eta)>$
$\displaystyle\hskip 5.69054pt\vdots$ $\displaystyle J_{r,i}$
$\displaystyle=J_{r-1,i}+<L_{g_{1}}\cdots
L_{g_{1}}v_{i_{i}}(\xi)-L_{g_{1}}\cdots L_{g_{1}}v_{i_{1}}(\eta),$
$\displaystyle\hskip 28.45274pt\cdots,L_{g_{m}}\cdots
L_{g_{m}}v_{i_{s_{i}}}(\xi)-L_{g_{m}}\cdots L_{g_{m}}v_{i_{s_{i}}}(\eta)>$

where $i_{1}:=\sum_{c=1}^{i-1}w_{c}+i-1$ and
$i_{s_{i}}:=\sum_{c=0}^{i}w_{c}+i-1$.

Further, define $J_{r}:=\sum_{i=1}^{q}J_{r,i}$. It holds that
$J_{0}\subset{J_{1}}\subset{J_{2}}\subset\cdots$. According to Hilbert’s basis
theorem and the ascending chain condition [19], there exists a certain ideal
satisfying

$\displaystyle J_{0}\subset{J_{1}}\subset{J_{2}}\subset\cdots\subset
J_{N}=J_{N+1}=\cdots=J$ (44)

for a certain number $N$. Define
$\phi:\mathbf{R}[\xi,\eta]\rightarrow\mathbf{R}[\xi]$ as the mapping that
substitutes a known initial state vector $\sigma$ into $\eta$. Denote the
ideal $\mathcal{J}\subset\mathbf{R}[\xi]$ as the image $\phi(J)$, which
characterizes the set of initial states that are indistinguishable from
$\sigma$. $\mathcal{J}$ is generated by a finite set of polynomials and its
algebraic variety $\mathbf{V}(\mathcal{J})$ consists of the common zeros of
those finite polynomials.

To help test the observability of the initial state
$\sigma=:(\sigma_{1},\sigma_{2},\cdots,\sigma_{n})^{T}$ for the higher-order
system (9), define the following ideal $\ell\subset\mathbf{R}[\xi]$ as

$\displaystyle\ell=:<\xi_{1}-\sigma_{1},\xi_{2}-\sigma_{2},\cdots,\xi_{n}-\sigma_{n},>$
(45)

Note that its algebraic variety $\mathbf{V}(\ell)=\\{\sigma\\}$. Then the
following theorem can be obtained.

Corollary 1 : The higher-order system (9) is globally observable at an initial
state $\xi\in\mathbf{R}^{n}$ if and only if one of the following equivalent
conditions holds

$\displaystyle(a)\ \mathbf{V}(\mathcal{J})=\mathbf{V}(\ell)$
$\displaystyle(b)\ \mathbf{V}(\sqrt{\mathcal{J}}:\ell)\subset\mathbf{V}(\ell)$

Proof :  For condition (a): Since $\mathbf{V}(\ell)=\\{\sigma\\}$, condition
(a) is equivalent to $\mathbf{V}(\mathcal{J})=\\{\sigma\\}$. The variety
$\mathbf{V}(\mathcal{J})$ is the set of initial states that are
indistinguishable from $\sigma$ such that condition (a) is a necessary and
sufficient condition for global observability of $\sigma$ according to
Definition 4.

For condition (b): Since for any two ideals $J,I\subset\mathbf{R}[\xi]$, the
following properties hold [19]

$\displaystyle\mathbf{V}(\sqrt{J}:I)$
$\displaystyle\subset\mathbf{V}(\sqrt{J})$ (46)
$\displaystyle\mathbf{V}(\sqrt{J})$ $\displaystyle=\mathbf{V}({J})$ (47)
$\displaystyle\mathbf{V}(\sqrt{J})\setminus\mathbf{V}(I)$
$\displaystyle\subset\mathbf{V}(\sqrt{J}:I)$ (48)

Combining properties (46) and (47), one can deduce that
$\mathbf{V}(\sqrt{\mathcal{J}}:\ell)\subset\mathbf{V}(\sqrt{\mathcal{J}})=\mathbf{V}(\mathcal{J})$.
This means that condition (b) implies condition (a). Inversely, according to
properties (47) and (48), it holds that

$\displaystyle\mathbf{V}(\mathcal{J})\setminus\mathbf{V}(\ell)=\mathbf{V}(\sqrt{\mathcal{J}})\setminus\mathbf{V}(\ell)\subset\mathbf{V}(\sqrt{\mathcal{J}}:\ell)$

If condition (b) holds, it yields that

$\displaystyle\mathbf{V}(\mathcal{J})\setminus\mathbf{V}(\ell)\subset\mathbf{V}(\ell)$

This leads to $\mathbf{V}(\mathcal{J})=\mathbf{V}(\ell)$. Therefore, condition
(b) implies condition (a) as well. This completes the proof. $\Box$

Example : Consider the higher-order system

$\displaystyle\dot{x}$
$\displaystyle=\left(\begin{array}[]{ccc}x_{2}x_{3}+x_{2}\\\
x_{1}x_{3}+x_{1}\\\ x_{1}x_{2}\end{array}\right)$ (52) $\displaystyle y$
$\displaystyle=x_{1}x_{2}x_{3}+x_{1}x_{2}$ (53)

where $x:=(x_{1},x_{2},x_{3})^{T}$. For the initial state pair $(\xi,\eta)$,
one can obtain that

$\displaystyle J_{0}$
$\displaystyle=<\xi_{1}\xi_{2}(\xi_{3}+1)-\eta_{1}\eta_{2}(\eta_{3}+1)>$
$\displaystyle J_{1}$
$\displaystyle=J_{0}+<(\xi_{2}\xi_{3}+\xi_{2})^{2}+(\xi_{1}\xi_{3}+\xi_{1})^{2}+\xi_{1}^{2}\xi_{2}^{2}$
$\displaystyle\hskip
14.22636pt-\left((\eta_{2}\eta_{3}+\eta_{2})^{2}+(\eta_{1}\eta_{3}+\eta_{1})^{2}+\eta_{1}^{2}\eta_{2}^{2}\right)>$
$\displaystyle J_{2}$
$\displaystyle=J_{1}+<2\xi_{1}\xi_{2}\left((\xi_{3}+1)^{2}+\xi_{2}^{2}\right)(\xi_{3}+1)$
$\displaystyle\hskip
14.22636pt+2\xi_{1}\xi_{2}\left((\xi_{3}+1)^{2}+\xi_{1}^{2}\right)(\xi_{3}+1)$
$\displaystyle\hskip
14.22636pt+\big{(}2(\xi_{2}\xi_{3}+\xi_{2})\xi_{2}+2(\xi_{1}\xi_{3}+\xi_{1})\xi_{1}\big{)}\xi_{1}\xi_{2}$
$\displaystyle\hskip
14.22636pt+2(\xi_{3}+1)\xi_{1}\xi_{2}(\xi_{1}^{2}+\xi_{2}^{2})-\big{(}2\eta_{1}\eta_{2}\left((\eta_{3}+1)^{2}+\eta_{2}^{2}\right)$
$\displaystyle\hskip
14.22636pt\times(\eta_{3}+1)+2\eta_{1}\eta_{2}\left((\eta_{3}+1)^{2}+\eta_{1}^{2}\right)(\eta_{3}+1)$
$\displaystyle\hskip
14.22636pt+\left(2(\eta_{2}\eta_{3}+\eta_{2})\eta_{2}+2(\eta_{1}\eta_{3}+\eta_{1})\eta_{1}\right)\eta_{1}\eta_{2}$
$\displaystyle\hskip
14.22636pt+2(\eta_{3}+1)\eta_{1}\eta_{2}(\eta_{1}^{2}+\eta_{2}^{2})\big{)}>$
$\displaystyle=J_{1}+<\xi_{1}\xi_{2}(\xi_{3}+1)\left(4(\xi_{3}+1)^{2}+6\xi_{1}^{2}+6\xi_{2}^{2}\right)$
$\displaystyle\hskip
14.22636pt-\left(\eta_{1}\eta_{2}(\eta_{3}+1)\left(4(\eta_{3}+1)^{2}+6\eta_{1}^{2}+6\eta_{2}^{2}\right)\right)>$

Therefore, it holds that

$\displaystyle J_{0}\subset{J_{1}}=J_{2}=J$ (54)

For any fixed initial state $\eta=\sigma$, according to Corollary 1, it yields
that

$\displaystyle\mathcal{J}=$
$\displaystyle<\xi_{1}\xi_{2}(\xi_{3}+1)-\sigma_{1}\sigma_{2}(\sigma_{3}+1),(\xi_{2}\xi_{3}+\xi_{2})^{2}$
$\displaystyle\hskip
14.22636pt+(\xi_{1}\xi_{3}+\xi_{1})^{2}+\xi_{1}^{2}\xi_{2}^{2}-(\sigma_{2}\sigma_{3}+\sigma_{2})^{2}$
$\displaystyle\hskip
14.22636pt-(\sigma_{1}\sigma_{3}+\sigma_{1})^{2}-\sigma_{1}^{2}\sigma_{2}^{2}>$
(55)

Based on the theory of solutions for equations with multiple varieties, the
set $\mathbf{V}(\mathcal{J})$ has multiple different elements. This means that
there always exists element $\xi\neq\sigma$ in set $\mathbf{V}(\mathcal{J})$
such that $\mathbf{V}(\mathcal{J})\neq\mathbf{V}(\ell)$. Then according to
Theorem 1, any initial state of the system (52)-(53) is undistinguished.
$\Box$

Based on the Gröbner basis and Hilbert’s Nullstellensatz theories [19], the
following sufficient conditions can be deduced by Corollary 1.

Corollary 2 : The higher-order system (9) is globally observable at an initial
state $\xi\in\mathbf{R}^{n}$ if one of the following conditions holds

$\displaystyle(a)\mathbf{V}(\sqrt{\mathcal{J}}:\ell)=\emptyset$
$\displaystyle(b)\mathcal{J}=\ell$
$\displaystyle(c)\sqrt{\mathcal{J}}:\ell=\mathbf{R}[\xi]$
$\displaystyle(d)1\in\sqrt{\mathcal{J}}:\ell$

$\Box$

Remark 2 : By transfer higher-order systems into polynomial forms, conditions
(a)-(b) in Corollary 2 are necessary and sufficient conditions to test global
observability of higher-order systems (9). The basic idea is to check the
common zeros of a finite set of equations given by generators of ideal
$\mathcal{J}$ or $(\sqrt{\mathcal{J}}:\ell)$. By virtue of the properties of
the polynomial ring and Hilbert’s basis theorem, the global observability
criteria are testable in finite steps different from the condition in Lemma 2,
which is difficult to determine how many Lie derivatives should be calculated.
Conditions (a)-(d) in Corollary 2 are sufficient conditions for testing global
observability of higher-order system (9), which is easier to verify comparing
with those in Corollary 1. Conditions (c)-(d) can be verified using Grobner
basis. $\Box$

###  III-C Local observability

Consider the integrated form (27) of a higher-order interacted system (9). The
local observability of the system (27) is determined by the dimension of the
space spanned by gradients of the Lie derivatives

$\displaystyle
L:=\sum_{i=1}^{n}f_{i}\frac{\partial}{\partial{x_{i}}}+\sum_{j=1}^{N-1}\sum_{l=1}^{w_{i}}u_{l}^{(j+1)}\frac{\partial}{\partial{u_{l}^{(j)}}}$
(56)

of its output function $h_{i}(x,u)$.

Define the observable Jacobian matrix of system (27) as

$\displaystyle
J=\left[\begin{array}[]{cccc}\frac{\partial{L_{f}^{0}h_{1}}}{\partial{x_{1}}}&\frac{\partial
L_{f}^{0}h_{1}}{\partial x_{2}}&\cdots&\frac{\partial L_{f}^{0}h_{1}}{\partial
x_{n}}\\\ \frac{\partial L_{f}^{0}h_{2}}{\partial x_{1}}&\frac{\partial
L_{f}^{0}h_{2}}{\partial x_{2}}&\cdots&\frac{\partial L_{f}^{0}h_{2}}{\partial
x_{n}}\\\ \vdots&\vdots&\vdots&\vdots\\\ \frac{\partial
L_{f}^{0}h_{q}}{\partial x_{1}}&\frac{\partial L_{f}^{0}h_{q}}{\partial
x_{2}}&\cdots&\frac{\partial L_{f}^{0}h_{q}}{\partial x_{n}}\\\ \frac{\partial
L_{f}^{1}h_{1}}{\partial x_{1}}&\frac{\partial L_{f}^{1}h_{1}}{\partial
x_{2}}&\cdots&\frac{\partial L_{f}^{1}h_{1}}{\partial x_{n}}\\\
\vdots&\vdots&\vdots&\vdots\\\ \frac{\partial L_{f}^{1}h_{q}}{\partial
x_{1}}&\frac{\partial L_{f}^{1}h_{q}}{\partial x_{2}}&\cdots&\frac{\partial
L_{f}^{1}h_{q}}{\partial x_{n}}\\\ \vdots&\vdots&\vdots&\vdots\\\
\frac{\partial L_{f}^{n-1}h_{q}}{\partial x_{1}}&\frac{\partial
L_{f}^{n-1}h_{q}}{\partial x_{2}}&\cdots&\frac{\partial
L_{f}^{n-1}h_{q}}{\partial x_{n}}\end{array}\right]_{nq\times n}$ (66)

The following local observability criterion can be directly obtained from the
existing results of observability for nonlinear systems [20][21][22].

Lemma 3 (local observability) :  The higher-order interacted system (9) is
locally observable if and only of the Jacobian matrix $J$ satisfies that
$\mathrm{rank}(J)=n$. $\Box$

First, consider the higher-order system (9) without input. According to (24),
the Kronecker form can be written as

$\displaystyle\begin{array}[]{cc}\dot{{x}}=&\sum_{k=1}^{c}\mathcal{A}_{k}{x}^{[k-1]},\
\ {x}(0)={x}_{o}\\\ y_{i}=&\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]},\ \
i=1,\cdots,q,\end{array}$ (69)

where $\mathcal{A}$ and $\mathcal{C}$ are defined in (24).

According to the property of products of matrices that $(A\otimes B)(C\otimes
D)=(AB)\otimes(CD)$, it holds that

$\displaystyle{L_{f}^{0}h_{i}}$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]}$
$\displaystyle{L_{f}^{1}h_{i}}$
$\displaystyle=\frac{d}{dt}\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]}=\sum_{k=1}^{c}\mathcal{C}_{i,k}\frac{d}{dt}\left(\overbrace{x\otimes
x\otimes\cdots\otimes x}^{k\ \mathrm{times}}\right)$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}\bigg{(}\dot{x}\otimes\cdots\otimes
x+\cdots+x\otimes\cdots\otimes\dot{x}\bigg{)}$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}\bigg{(}\sum_{m=2}^{c}\mathcal{A}_{m}{x}^{[m-1]}\otimes\cdots\otimes
x+\cdots$ $\displaystyle\hskip
14.22636pt+x\otimes\cdots\otimes\sum_{m=2}^{c}\mathcal{A}_{m}{x}^{[m-1]}\bigg{)}$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}\sum_{j=1}^{k}\left(x\otimes\cdots\otimes\underbrace{\sum_{m=2}^{c}\mathcal{A}_{m}{x}^{[m-1]}}_{j-th\
\mathrm{position}}\otimes\cdots\otimes x\right)$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}\sum_{j=1}^{k}\sum_{m=2}^{c}\left(I\otimes\cdots\otimes\underbrace{\mathcal{A}_{m}}_{j-th\
\mathrm{position}}\otimes\cdots\otimes I\right)$ $\displaystyle\hskip
14.22636pt\times{x}^{[k+m-2]}$ $\displaystyle\vdots$
$\displaystyle{L_{f}^{n}h_{i}}$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}\bar{A}_{1}\bar{A}_{2}\cdots\bar{A}_{n}x^{[k+n(m-2)]}$

where

$\displaystyle\bar{A}_{n}=\sum_{j=1}^{k+(n-1)(m-2)}\sum_{m=2}^{c}\left(\overbrace{I\otimes\cdots\otimes\underbrace{\mathcal{A}_{m}}_{j-th\
\mathrm{position}}\otimes\cdots\otimes I}^{k+(n-1)(m-2)\
\mathrm{times}}\right)$ (70)

Then, the observability matrix $\mathbf{O}(x)$ is

$\displaystyle\mathbf{O}(x)=\nabla_{x}\left(\begin{array}[]{cccc}\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]}\\\
\sum_{k=1}^{c}\mathcal{C}_{i,k}\bar{A}_{1}{x}^{[k+m-2]}\\\ \vdots\\\
\sum_{k=1}^{c}\mathcal{C}_{i,k}\bar{A}_{1}\bar{A}_{2}\cdots\bar{A}_{n}x^{[k+n(m-2)]}\end{array}\right)$
(75)

Then, according to Lemma 3, the following theorem holds.

Theorem 1 : The higher-order system

$\displaystyle\dot{{x}}$
$\displaystyle=\sum_{k=1}^{c}\mathbf{A}_{k}{x}^{k-1},$ $\displaystyle y_{i}$
$\displaystyle=\sum_{k=1}^{c}\mathbf{C}_{i,k}{x}^{k}$

is locally observable if and only of $\text{rank}(\mathbf{O}(x))=n$. $\Box$

Now we consider the local observability of the higher-order interacted system
(9) with input $\mathbf{D}$. Based on (56) and the Kronecker form (24), it
yields that

$\displaystyle{L_{f}^{0}h_{i}}$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}{x}^{[k]}u_{l}$
$\displaystyle{L_{f}^{1}h_{i}}$
$\displaystyle=\sum_{k=1}^{c}\mathcal{C}_{i,k}\frac{d{x}^{[k]}}{dt}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}\frac{d{x}^{[k]}}{dt}u_{l}$
$\displaystyle\hskip
14.22636pt+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}{x}^{[k]}\frac{du_{l}}{dt}$
$\displaystyle=\left(\sum_{k=1}^{c}\mathcal{C}_{i,k}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}u_{l}\right)\sum_{m=2}^{c}\sum_{j=1}^{k}$
$\displaystyle\hskip
14.22636pt\left(I\otimes\cdots\otimes\underbrace{\mathcal{A}_{m}}_{j-th\
\mathrm{position}}\otimes\cdots\otimes I\right){x}^{[k+m-2]}$ $\displaystyle\
\ \ \
+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}{x}^{[k]}\frac{du_{l}}{dt}$
$\displaystyle\hskip 8.53581pt\vdots$ $\displaystyle{L_{f}^{n}h_{i}}$
$\displaystyle=\\!a_{1}\\!\\!\left(\sum_{k=1}^{c}\mathcal{C}_{i,k}\\!+\\!\\!\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}u_{l}\\!\right)\\!\\!\bar{A}_{1}\bar{A}_{2}\cdots\bar{A}_{n}x^{[k+n(m-2)]}$
$\displaystyle\ \ \ \
+a_{2}\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}\bar{A}_{1}\cdots\bar{A}_{n-1}x^{[k+(n-1)(m-2)]}\frac{du}{dt}$
$\displaystyle\ \ \ \
+a_{3}\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}\bar{A}_{1}\cdots\bar{A}_{n-2}x^{[k+(n-2)(m-2)]}\frac{d^{2}u}{dt}$
$\displaystyle\ \ \ \
+\cdots+a_{n}\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}\bar{A}_{1}x^{[k]}\frac{d^{n}u}{dt}$

where $\bar{A}_{n}$ is defined in (70). $a_{1},a_{2},\cdots,a_{n}$ correspond
to the terms in the $n$-th raw of the Yanghui triangle in turn. According to
Yanghui triangle’s property, they are equal to the coefficients in the
expansion of $(a+b)^{n-1}$.

Then, the observability matrix $\mathbf{O}_{1}(x)$ is written as

$\displaystyle\mathbf{O}_{1}(x)=\nabla_{x}\left(\begin{array}[]{ccccc}\sum_{k=1}^{c}\mathcal{C}_{i,k}{x}^{[k]}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}{x}^{[k]}u_{l}\\\
\left(\sum_{k=1}^{c}\mathcal{C}_{i,k}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}u_{l}\right)\sum_{m=2}^{c}\sum_{j=1}^{k}\bar{A}_{1}{x}^{[k+m-2]}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}{x}^{[k]}\frac{du_{l}}{dt}\\\
\vdots\\\
\left(\sum_{k=1}^{c}\mathcal{C}_{i,k}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}u_{l}\right)\bar{A}_{1}\bar{A}_{2}\cdots\bar{A}_{n}x^{[k+n(m-2)]}\\\
+\sum_{p=1}^{n-1}a_{p+1}\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathcal{D}_{i,k,l}\bar{A}_{1}\cdots\bar{A}_{n-p}x^{[k+(n-p)(m-2)]}\frac{d^{p}u}{dt}\end{array}\right)$

Furthermore, according to Lemma 3, the following result holds.

Theorem 2 : The higher-order system

$\displaystyle\dot{{x}}$
$\displaystyle=\sum_{k=1}^{c}\mathbf{A}_{k}{x}^{k-1},$ $\displaystyle y_{i}$
$\displaystyle=\sum_{k=1}^{c}\mathbf{C}_{i,k}{x}^{k}+\sum_{l=1}^{w_{i}}\sum_{k=1}^{c}\mathbf{D}_{i,k,l}{x}^{k}u_{l}$

is locally observable if and only of $\text{rank}(\mathbf{O}_{1}(x))=n$.
$\Box$

###  III-D Similarity of observability

To figure out the relation of observability between two different hypergraphs,
based on the similarity of tensors, we consider the similarity of
observability for higher-order systems.

Towards this end, let us first introduce the concept of similarity of tensors.

Definition 6 (Similarity of tensor [23]) : Let $\mathbf{A}$ and $\mathbf{B}$
be two $m$-order $n$-dimension tensors. Tensors $\mathbf{A}$ and $\mathbf{B}$
are diagonal-similar if there exists some $n$-order invertible diagonal matrix
$D$ such that $\mathbf{B}=D^{-(m-1)}\mathbf{A}D$.

Lemma 4 (Similarity of observability) : If the higher-order system
$\dot{{x}}=\mathbf{A}{x}^{k-1}$, where $A$ is an even-order tensor, with
output $y=g({x})$ is globally observable at ${x}_{0}=\varepsilon$, then the
higher-order system $\dot{{z}}=\mathbf{\bar{A}}{z}^{k-1}$, where
$\mathbf{\bar{A}}=R^{1-m}\mathbf{A}R$ and
$R\in\mathbf{R}^{n}\times\mathbf{R}^{n}$ is a diagonal matrix satisfying
$R_{ii}=\pm 1,\forall i,j\in\\{1,2,\cdots,n\\}$, with output
$y^{\prime}=g(R^{-1}{z})$ is globally observable at ${z}_{0}=R\varepsilon$.

Proof :  Consider a higher-order system

$\displaystyle\dot{{x}}$ $\displaystyle=\mathbf{A}{x}^{k-1}$ $\displaystyle y$
$\displaystyle=h({x})$

Under the coordinate transformation ${z}=R{x}$, the system transfers into

$\displaystyle\dot{{z}}$ $\displaystyle=R\mathbf{A}(R^{-1}{z})^{k-1}$

Component-wise, it holds that

$\displaystyle\dot{z}_{i}$
$\displaystyle=R_{ii}\sum_{i_{2},i_{3},\cdot,i_{k}=1}^{n}A_{ii_{2}i_{3}\cdots
i_{k}}R^{-1}_{i_{2}i_{2}}z_{i_{2}}R^{-1}_{i_{3}i_{3}}z_{i_{3}}\cdots
R^{-1}_{i_{m}i_{m}}z_{i_{m}}$
$\displaystyle=\sum_{i_{2},i_{3},\cdot,i_{k}=1}^{n}A_{ii_{2}i_{3}\cdots
i_{k}}R_{ii}R^{-1}_{i_{2}i_{2}}R^{-1}_{i_{3}i_{3}}\cdots
R^{-1}_{i_{m}i_{m}}z_{i_{2}}z_{i_{3}}\cdots z_{i_{m}}$

Since $m$ is even, compared with $\mathbf{B}=D^{-(m-1)}\mathbf{A}D$, we can
directly use $R$ as $D$ to construct a similar tensor with the fact
$R_{ii}=R_{ii}^{1-m}$.

Therefore,

$\displaystyle\dot{{z}}$ $\displaystyle=\mathbf{\bar{A}}{z}^{k-1}.$

Thus, $L_{\tau_{1}}\cdots
L_{\tau_{r}}v_{i}(x)=dL_{\tau_{j}}v_{i}(z),i=1,\cdots,q,r=0,1,\cdots$ where
$\tau_{1},\cdots,\tau_{r}\in\\{g_{0},\cdots,g_{m}\\}$ and
$v_{i}\in\\{p_{i,0},p_{i,m}\\}$ Then according to Lemma 2, the observability
of transformed system $\dot{z}=\bar{\mathbf{A}}z$. $\Box$

##  IV Conclusion

In this paper, we consider both global and local observability of higher-order
systems with both inputs and outputs by exploiting algebraic conditions and
rank conditions, respectively. Based on the properties of polynomial ideals
and varieties in commutative algebra, global observability criteria of
hypergraphs are established. An example is provided to prove the effectiveness
of the criteria. On the other hand, local observability criteria are obtained
based on the rank condition of the observability matrix. Finally, to further
simplify the testing times, the similarity of observability is proposed based
on the similarity of tensors.

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11institutetext:  ✉Yupeng Xu
11email<EMAIL_ADDRESS>
✉Yusa Wang
11email<EMAIL_ADDRESS>
22institutetext: 1 Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China.
33institutetext: 2 University of Chinese Academy of Sciences, Chinese Academy
of Sciences, Beijing 100049, China. 44institutetext: 3 Physics and space
science college, China West Normal University, Sichuan, 637002, China
55institutetext: 4 College of Physics, Jilin University, Changchun 130012,
China

# Design and test of a portable Gamma-Ray Burst simulator for GECAM

Can Chen 1,2 Shuo Xiao 1,2 Shaolin Xiong 1 Nian Yu 1,2 Xiangyang Wen 1 Ke Gong
1 Xinqiao Li 1 Chaoyang Li 1,3 Dongjie Hou 1 Xiongtao Yang 1 Zijian Zhao 1
Yuxuan Zhu 1, 4 Dali Zhang 1 Zhenghua An 1 Xiaoyun Zhao 1 Yupeng Xu 1,2,* Yusa
Wang 1,*

(Received: date / Accepted: date)

###### Abstract

The main scientific goal of the Gravitational wave high-energy Electromagnetic
Counterpart All-sky Monitor (GECAM) is to monitor various types of Gamma-Ray
Bursts (GRB) originated from merger of binary compact stars, which could also
produce gravitational wave, and collapse of massive stars. In order to study
the response of GECAM Gamma-Ray Detectors (GRDs) to high-energy bursts and
test the in-flight trigger and localization software of GECAM before the
launch, a portable GRB simulator device is designed and implemented based on
grid controlled X-ray tube (GCXT) and direct digital synthesis (DDS)
technologies. The design of this GRB simulator which modulates X-ray flux
powered by high voltage up to 20 kV is demonstrated, and the time jitter
(FWHM) of the device is about 0.9 $\upmu$s. Before the launch in December,
2020, both two GECAM satellites were irradiated by different types of GRBs
(including short and long bursts in duration) generated by this GRB simulator.
The light curves detected with GECAM/GRDs are consistent with the programmed
input functions within statistical uncertainties, indicating the good
performance of both the GRDs and the GRB simulator.

###### Keywords:

GECAM Gamma-ray burst Grid controlled X-ray tube Simulator

††journal: Experimental Astronomy

##  1 Introduction

Gamma-Ray Burst (GRB), the most violent explosion after the Big Bang [1], is a
phenomenon that gamma-rays from the distant universe suddenly increase in a
short time scale. GRBs are usually divided into long bursts or short bursts
according to the duration of $\rm T_{90}$ which is the time interval between
5% and 95% of the cumulative observed fluence above the background [2]. Short
bursts are believed to originate from binary compact star mergers, which has
been firmly demonstrated by the first binary neutron star merger (GW 170817)
and its high-energy electromagnetic counterpart (GRB 170817A) [3][4], while
long bursts are produced by collapse explosions of massive stars.

To monitor the high-energy electromagnetic radiation of gravitational waves
and other gamma-ray bursts, Gravitational wave high-energy Electromagnetic
Counterpart All-sky Monitor (GECAM) was proposed in 2016. GECAM is composed of
two microsatellites[5], and each GECAM satellite contains 25 Gamma-Ray
Detectors (GRDs) and 8 Charged Particle Detectors (CPDs). The illustration of
a GECAM satellite is shown in Fig. 1 (left). As the main scientific payload,
GRD is implemented by coupling $\rm LaBr_{3}$:Ce with the Silicon
Photomultiplier (SiPM) array, and the geometry area of each GRD is about 45
$\rm cm^{2}$. The photo of GRD is shown in Fig. 1 (right). Each GRD is
equipped with a high gain and a low gain signal processing channel [6][7] to
achieve a wide dynamic detection range (about 6 to 5000 keV).

Figure 1: Left: Each GECAM satellite is equipped with 25 GRDs and 8 CPDs.
Right: The photo of a GRD of GECAM.

Response of detectors to different type of GRBs could be studied with
numerical simulations based Monte Carlo modeling, as for example, it has been
done for the Gamma-Ray Burst Monitor (GBM), one of the scientific payloads of
the Fermi [8]. However, to study the real behavior of GECAM/GRD to GRBs and
verify the in-flight trigger and localization software (X.Y. Zhao et al. In
preparation), an X-ray source with programmable light curves and intensities
to mimic various gamma-ray bursts on-ground is required.

Stroboscopic X-ray generators based on condenser-discharge induced flash X-ray
technique were designed and used for X-ray imaging in 1999 [9], however, this
method cannot produce X-rays with continuous varying photon intensity. Pulse
X-ray source based on chopper was designed in 2011 [10], but its turntables
need to be replaced if one wants to produce a different light curve, which
leads to poor flexibility.

In this study, the GRB simulator was designed based on the grid controlled
X-ray tube (GCXT) which had been used for X-ray communication and X-ray
pulsar-based navigation simulation source [11][12]. Based on the traditional
X-ray tube, a grid electrode is added between the anode and the cathode to
control the flux of electrons bombarding the anode by applying a programmable
negative voltage vs the cathode. Thus, a GCXT can generate high-energy photons
with variable intensity. As in a triode system, the grid cut-off voltage
determining the state of negligible tube current is directly related to the
anode voltage and so to the maximum X-ray energy. Due to the modulation
voltage swing, the energy modulation range is limited to the soft X-ray band
(about 1 to 10 keV) [13], which is relatively lower than the observation
energy range (about 6 keV to 5 MeV) of GECAM [14]. A newly released high-
voltage precision operational amplifier ADHV4702-1 manufactured by Analog
Devices Inc. [15] is applied in the GRB simulator to achieve the modulation of
X-ray at higher energies (up to about 20 keV). In addition, considering the
complexity and irregularity of actual GRB light curve, a method of directly
synthesizing arbitrary waveforms called direct digital synthesis (DDS)[16] is
adopted in making a modulation voltage signal source.

This paper is structured as follows. The design of the GRB simulator is
introduced in Section 2. Next, the performance of the GRB simulator is
presented in Section 3. Then, the GRB ground tests with the GRB simulator
performed before the launch of GECAM is depicted in Section 4. Finally, the
performance of the GRB simulator and the response of GRDs to GRBs are
summarized in Section 5.

##  2 Design of the GRB simulator

The GRB simulator is mainly composed of two parts: DDS modulator and GCXT.
Each part is modularized to make the device portable and convenient to
upgrade, as shown in Fig. 2. The firmware containing GRB light curve data is
loaded into the Field-Programmable Gate Array (FPGA) inside the DDS modulator
through the computer. Then the DDS modulator generates a modulation signal
transmitted to the grid electrode of GCXT. Finally, GCXT produces the
corresponding X-ray flux following the input light curve.

Figure 2: Schematic of the GRB simulator mainly consist of DDS modulator and
GCXT.

###  2.1 DDS modulator

DDS, a technique of directly synthesizing analog waveform, is utilized to
generates artificial GRBs with arbitrary light curves. The DDS modulator is
mainly realized with three evaluation boards, as shown in Fig. 3, a controller
based on Altera EP4CE6E22C8 FPGA, a Digital-to-Analog Converter (DAC) board
consist of an ADI 14-bit AD9767 DAC with parallel interface to the FPGA, and a
driver module with a high voltage op-amp ADHV4702-1.

In order to accurately reproduce a light curve with timescale of the shortest
GRBs (about 2 ms) [17], a stable global clock with frequency of $f_{\rm
clk}=100\,{\rm MHz}$ inside the FPGA is generated with PLL (Phase Locked Loop)
frequency multiplication through the on-board $\rm 50\,MHz$ crystal
oscillator. The output frequency of a light curve phase point is managed with
a $N$-bit phase accumulator register and a frequency control word register
with the value of $K$ (ranging from 1 to $2^{N}-1$). The phase accumulator
accumulates continuously with the addition of $K$ at frequency $f_{\rm clk}$.
A phase point is determined and the accumulator resets to zero when it
overflows. Thus, the output frequency is the rate of the phase accumulator
overflows [16]:

$f_{\rm out}=f_{\rm clk}\cdot K/2^{N}$ (1)

where the value of $N$ determines the lowest phase clock could be obtained and
is fixed to 32 in this study.

The data describing the GRB light curve, stored in the look up table (LUT),
are sent to the DAC through the parallel interface at each phase point, i.e.
with frequencies of $f_{\rm out}$. Therefore, a nearly continuously adjustable
output frequency could be controlled by the single parameter $K$ and the
duration of the light curve, i.e. the number of phase points, is limited by
the memory size of the LUT.

To drive the GCXT, a high voltage amplifier is essential. Considering the
future requirement to achieve higher energy X-ray modulation (at least 50 keV)
and extendibility, ADHV4702-1 is used to amplify the modulation signal from
the DAC. ADHV4702-1 is the first 220 V high-voltage precision operational
amplifier with a wide dynamic range (-110 V$\sim$+110 V) and a relatively high
slew rate (74V/µs typical) [15].

Figure 3: Upper: Simplified schematic of DDS modulator. Lower-left: Circuit
board of FPGA and DAC. Lower-right: ADI ADHV4702-1 high-voltage precision
operational amplifier board whose clamp diode is removed to maximizes slew
rate.  Figure 4: Simplified schematic of the GCXT, mainly composed of cathode,
grid electrode and anode.

###  2.2 Design of grid controlled X-ray tube

The GCXT we designed is in a cylindrical shape with a height of 17 cm and a
diameter of 8 cm. The simplified schematic of GCXT is shown in Fig. 4. Based
on the classical X-ray tube, a grid electrode with programmable negative
voltage is added between the cathode and the anode. The tungsten cathode
filament is heated by a controllable current source to emit thermal electrons
with intensity proportional to the filament current. The electric field
produced by the high voltage applied to the anode accelerates the thermal-
emitted electrons that bombarding the anode target generates X-rays by
bremsstrahlung including the characteristic lines of the anode material. Due
to the thermal inertia, the output intensity can not be controlled by the
filament current for fast timing signal simulation. Instead, a negative
voltage applied to the grid electrode close to the cathode could control the
electron flux passing through promptly, and therefore, control the intensity
of the X-ray generated by the X-ray tube.

##  3 Performance of the GRB simulator

###  3.1 Grid voltage modulation properties of the GRB simulator

To verify the performance of modulating X-ray flux generated by the GCXT, a
KETEK AXAS-D silicon drift detector (SDD) is used to measure X-ray flux and
spectrum emitted from the GCXT under different grid voltages. The experimental
layout is shown in Fig. 5 (left). The SDD with 30 $\rm mm^{2}$ active area is
placed about 5 cm from the GCXT. The anode voltage is set to +20 kV supplied
by the Wisman High voltage source XEL20P, while the voltage of the cathode is
fixed to 0 V.

The detected raw energy distribution is shown in Fig. 6. Grid electrode
voltage is set from 0 V to -20 V with a step size of 1 V to investigate the
voltage response of GCXT, and the recording time of each voltage point is 100
s. The relationship between the count rates detected by SDD and the grid
electrode voltage is shown in Fig. 5 (right). The upper threshold of grid
electrode voltage to detect X-ray is about -18 V. We find that there is a
relatively linear relationship between grid electrode voltage and output count
rates in the range of -15 V to -8 V. The least squares method is adopted to
testify the relationship between two variables:

$C=1236.7\cdot(V_{\rm grid}+15.4)$ (2)

where $C$ is the count rates (counts per second) acquired by SDD, $V_{\rm
grid}$ is the voltage of grid electrode. Meanwhile, photon count rates
increase with filament current, while maintaining the linear relationship in
this voltage range. The GRB simulator works in this linear region during the
following experiment to GECAM.

Thus, a specific GRB light curve can be generated by setting the corresponding
grid electrode voltage at each phase point of a light curve.

Figure 5: Left: Experimental layout to investigate the modulation properties
of GCXT. Right: Output count rates of GCXT versus grid electrode voltage.
Figure 6: The raw energy distribution of GCXT detected by KETEK SDD when the
high voltage source is set to 20 kV. The characteristic emission line of
tungsten can be clearly seen, including the 8.40 keV W-$\rm L_{\upalpha 1}$
line and the 9.67 keV W-$\rm L_{\upbeta 1}$ line.

###  3.2 Time characteristic of the GRB simulator

Time delay and time jitter between the application of the grid signals and the
X-ray produced are caused mainly during the process of electron drifting and
bombarding the anode, which directly affects the quality of GRB reproduced.
Therefore, the time response of the GRB simulator is also investigated in this
study.

The experimental schematic is shown in Fig. 7. The grid electrode of the GCXT
is triggered by a square pulse signal with a duration of 1 $\upmu$s and a 100
kHz repeating frequency generated by the DDS modulator. Simultaneously, the
same signal is sent to the Carmel NK732 time interval analyzer (TIA) as the
start signal. NK732 is able to time tag events (edges of an input pulse train)
at a rate of 20 million per second continuously with 2 ps resolution[18]. The
arrival time of the X-ray pulse indicated by the discriminator output of a
homemade SDD shaping circuit[19] with a peaking time of about 0.55 $\upmu$s is
used as the stop signal to the TIA. The distribution of time interval between
the raising edge of the square pulse signal applied to the grid electrode
(start signal) and arrival time of the first detected X-ray photon in the SDD
(stop signal) is shown in Fig. 8. The time jitter (Full Width at Half Maximum,
FWHM) of the GRB simulator equals 0.9 $\upmu$s, and the time delay is 1.6
$\upmu$s with respect to the start of signal applied to the grid electrode.
Therefore, the minimum variable time scale of GRB light curve generated by the
GRB simulator is 0.9 $\upmu$s. Taking into account the uncertain time
introduced by the SDD signal shaping as well as the spread of photon
generating time within square pulse, the time response of the GRB simulator
measured here is a relatively rough estimate (upper limit).

Figure 7: Simplified experimental schematic to investigate the time response
of GCXT. Figure 8: Time delay and time jitter of the GRB simulator. The blue
line represents the detected time interval. The red line is the result of
Gaussian fitting.

##  4 GRB simulation experiment for GECAM on the ground

Ground experiment to study the response to GRB and verify the in-flight
trigger and localization software has been carried out to the two GECAM
satellites (denoted as GECAM-01 and GECAM-02) comprehensively. The schematic
diagram of this experimental layout is shown in Fig. 9. Limited by the opening
angle of GCXT photon output window (about 20°), the GCXT is placed at the
perpendicular bisector of GECAM-01 and GECAM-02 with about 4 m away from these
two satellites to allow X-rays irradiating both of them at the same time. The
distance between the two satellites is about 1 m. Some GRDs (#5, #6, #13, #14,
#15, #16, #22, #23, #24)[14] facing to the X-ray tube can receive more X-ray
photons because they have larger receiving area to the X-ray flux. Considering
the maximum photon energy generated by GCXT is about 20 keV, only the high
gain channel of the GRD could detect these photons.

Figure 9: The layout of the ground GRB experimental to GECAM. GCXT is placed
at the perpendicular bisector of GECAM-01 and GECAM-02 with about 4 m away
from two satellites to allow X-rays to irradiating two satellites at the same
time. The distance between the center of the two satellites is 1 m.

GRBs are commonly described as fast-rise exponential-decay (FRED) shapes (e.g.
Fenimore et al. 1996)[20]. However, the actual GRB light curve is complex and
irregular. Hence, the shape of FRED and “super-long” curve synthesized with
several actual GRBs that observed by Fermi/GBM were reproduced through the GRB
simulator. In our tests, the duration of FRED shape GRB is set to 0.5 s, as a
result, $N$ and $K$ should be set to 32 and 858993 respectively according to
eq. (1). The theoretical curve is the expected photon count rates detected by
GRDs:

$C_{\rm theoretical}(t)=a\cdot f(V_{\rm grid}(t))$ (3)

where $a$ is a constant that relate to the response properties and effected
area of different detectors, $f$ is the relationship between the grid
electrode voltage and the output count rates of GCXT, as shown in eq. (2),
$V_{\rm grid}(t)$ is the grid electrode voltage of each phase point in a light
curve (i.e. the programmed input functions).

Since the start time of the simulated GRB generated by the GRB simulator is
artificially set to random, the Discrete Cross Correlation Function (DCCF)
method [21][22][23] is applied to align the detected light curves with the
theoretical light curves input to the GRB simulator on the time axis.
Additionally, we performed a chi-square fit between the detected light curves
and the theoretical curves (see eq. (3)) as well. The detected light curve and
theoretical FRED curve are shown in the upper of Fig. 10, and the residual
distribution of these two curves is shown in the bottom of Fig. 10. Since the
statistical fluctuations of the counts in each bin approximately follow the
Poisson distribution, and we define the residual:

$R=(C_{\rm detected}-C_{\rm theoretical})/{\sqrt{C_{\rm theoretical}}}$ (4)

where $C_{\rm detected}$, $C_{\rm theoretical}$ are the detected count rates,
the theoretical count rates, respectively. The mean and standard deviation
fitted with Gaussian of the residual distribution equals $-0.022\pm 0.052$,
$0.866\pm 0.040$ (with 1.019 “reduce-$\upchi^{2}$”), respectively. The
detected super-long light curve are shown in the upper of Fig. 11, and the
residual distribution of the two curves is shown in the bottom of Fig. 11. The
mean and standard deviation of the distribution equals $-0.006\pm 0.022$,
$1.039\pm 0.016$ (with 1.037 “reduce-$\upchi^{2}$”), respectively.

Moreover, the stability of photon energy spectrum over time is studied by
selecting two different adjacent time intervals in the super-long GRB light
curve illustrated in the upper of Fig. 11. Each time interval has a 30 seconds
in duration, and the total counts in the two intervals has a ratio of 1 to 2.
The energy spectra detected by GRDs in different time intervals are shown in
the upper panel of Fig. 12, while the ratio of the two spectra for each energy
bin is shown in the bottom panel of Fig. 12. It can be found that the photon
energy spectrum is consistent in different time intervals, which proved the
excellent energy stability of this GRB simulator we designed.

Figure 10: The upper panel is the FRED shape GRB light curve we constructed
(blue line), then reproduced by the GRB simulator and detected by GRDs (orange
line). GRD#5, #6, #13, #14, #15, #16, #22, #23 and #24 were used. The bottom
panel is the residual of the theoretical curve and the detected light curve.
Bottom: The distribution of the residual. The red line is the result of
Gaussian fitting.

Figure 11: Upper: The upper panel is the “super-long” curve synthesized with
several actual GRBs observed by Fermi/GBM (blue line), then reproduced by the
GRB simulator and detected by GRDs (orange line). GRD#5, #6, #13, #14, #15,
#16, #22, #23 and #24 were selected . The bottom panel is the residual of
theoretical curve and detected light curve. Bottom: The distribution of the
residual. The red line is the result of Gaussian fitting.  Figure 12: The
shape of the energy spectrum detected by GRD in a different time range of
super-long GBR (in Fig. 11) is consistent which illustrates the good energy
spectrum stability of the device over time.

##  5 Conclusion and discussion

In this work, a new GRB simulator based on grid controlled X-ray tube and
direct digital synthesis technical is designed and used to verify the
performance of the GECAM/GRD and the in-flight trigger and localization
software of GECAM to different kinds of GRBs during the ground tests before
the launch. Both the relationship of output count rates to different grid
electrode voltage and the time response of the GRB simulator were
investigated. The Characteristic of the GRB simulator is shown in Table 1,
which meet the requirement of the GRB simulation experiment for GECAM.

Table 1: Characteristic of the GRB generator Item | Value | Notes
---|---|---
Energy range | 3 - 20 keV |  | Can be extended by applying
---
higher voltage
Beam angle | About 20 ° | -
| Minimum
---
time variable scale
About 0.9 $\upmu$s | Rough estimated (upper limit)
| Maximum
---
duration time
| Theoretically
---
infinite
| The duration time divided by
---
the time binned of DDS module
is less than or equal to
16384 (size of ROM)
Output count rates | About 0 to 1 Mcps |  | Can be extended by
---
increasing filament current
Polarization | - |  | Polarized radiation
---
according to [24]

Several GRBs generated through this GRB simulator were used to test the GECAM
satellites in the configuration shown in Fig. 9. The results show that the
detected light curves are in good agreement with the input theoretical curves
within the statistical uncertainty, even when the time scale of FRED is very
short (with 1.6 ms time binned), demonstrating the ability of GECAM in GRB
timing studies[25][26] and in the triangulating GRBs by using the time
difference between the light curve detected by different satellites
[27][28][29]. In particular, we note that the residual uncertainty is
relatively large at some point with the highest brightness (i.e. the observed
counts is less than the theoretical count), which probably caused by the
effect of detector deadtime (approximately 4 $\upmu$s per event).

Furthermore, a 100-meter X-ray space telescope testing facility [30] has been
built by IHEP in Beijing, and the portable GRB simulator can be installed at
one end of the facility through the standard KF40 vacuum flange to obtain
quasi-parallel light at the other end of the facility without the requirement
of a collimator.

Considering the limitation of heat dissipation, the maximum voltage that could
be applied to the X-ray tube we designed is about 20 kV, which means that this
GRB simulator can only produce X-rays below about 20 keV (much softer than
normal GRBs). In our next work, water cooling will be added to the X-ray tube
to increase the rated voltage, achieving the modulate of higher X-ray energy
up to 50 keV or more, which will allow us to mimic GRBs with normal spectral
hardness.

## Acknowledgements

This research is supported by the Strategic Priority Program on Space Science,
the Chinese Academy of Sciences, Grant No. XDA15020501, No. XDA1531010301 and
No. XDA15360102. We are also grateful to Shanghai KEYWAY ELECTRON Technology
Co., Ltd for their assistance in the processing and design of X-ray tubes.

## Conflict of interest

The authors declare that they have no conflict of interest.

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