Explicit Characterization of Performance of a Class of Networked Linear Control Systems

TL;DR

This paper provides a unified framework to analyze the steady-state variance performance of networked linear control systems, linking it to Laplacian eigenvalues and offering insights into connectivity thresholds and scalability.

Contribution

It introduces a general method to characterize network performance via Laplacian eigenvalues, extending previous results to arbitrary nodal dynamics and observer-based feedback.

Findings

Performance expressed as sum over Laplacian eigenvalues

Derived bounds and scaling laws for network performance

Extended methodology to observer-based and composite networks

Abstract

We show that the steady-state variance as a performance measure for a class of networked linear control systems is expressible as the summation of a rational function over the Laplacian eigenvalues of the network graph. Moreover, we characterize the role of connectivity thresholds for the feedback (and observer) gain design of these networks. We use our framework to derive bounds and scaling laws for the performance of the dynamical network. Our approach generalizes and unifies the previous results on the performance measure of these networks for the case of arbitrary nodal dynamics. We bring extensions of our methodology for the case of decentralized observer-based output feedback as well as a class of composite networks. Numerous examples support our theoretical contributions.

Figures

Tables

Table 1. Table I: The subsystems investigated in Example 1 together with the performance functions in the case of relative state-feedback with σ = 0 𝜎 0 \sigma=0 . We assume that a , a 1 , a 2 ≥ 0 𝑎 subscript 𝑎 1 subscript 𝑎 2 0 a,a_{1},a_{2}\geq 0 .

Realization	$\varphi (\lambda ,𝐊)$
$\begin{array}{c}{𝔰}_1:\\ 𝐀=-a,\\ 𝐁=𝐄=1,𝐂=1\end{array}$	$\frac{1}{2\left(k\lambda +a\right)}$
	$\frac{b_0^2{k}_1\lambda +a_2{b}_0^2+b_1^2}{2\left(k_2\lambda +a_1\right)\left(k_1\lambda +a_2\right)}$

Table 2. Table II: Performance functions for a composite network with complete graph subnetworks of single and double-integrator agents. Each module has m 𝑚 m nodes and the feedback gains are assumed to be identical over both graphs (see Example 9 ).

Dynamics	Performance Function ${\varphi }_{\mathrm{nn}}\left(\lambda \right)$
single-integrator	$\frac{2\left(m-1\right)\lambda +m^2}{2mk\lambda }$
double-integrator	$\frac{\left(m-1\right)\left(m+2\right){\lambda }^2+2m^2\left(m-1\right)\lambda +m^4}{2m^2{k}_1{k}_2{\lambda }^2}$

Table 3. Table III: The value of 𝐏 ​ ( λ , 𝐊 ) 𝐏 𝜆 𝐊 {\mathbf{P}}(\lambda,\mathbf{K}) as the solution of Lyapunov equation for triple integrators.

Table 4. Table IV: The solution to Lyapunov equation 𝐏 ~ ~ 𝐏 \tilde{\mathbf{P}} for complete subnetworks with m 𝑚 m double-integrator agents ( ∗ * implies symmetric element).

Equations

$$\displaystyle l_{ij}=\left\{\begin{array}[]{ll}\displaystyle\sum_{\{i,j\}\in\mathcal{E}}k_{ij}&\mbox{if }i=j\\ \leavevmode\nobreak\ -k_{ij}&\mbox{if }i\neq j\end{array}\right..$$

$$\displaystyle S_{i}:\left\{\begin{array}[]{l}\dot{x}_{i}(t)\,=\,\mathbf{A}\,x_{i}(t)+{\mathbf{B}}\,u_{i}(t)+{\mathbf{E}}\,\xi_{i}(t)\\ y_{i}(t)\,=\,{\mathbf{H}}\,x_{i}(t)+\sigma\,\eta_{i}(t)\\ z_{i}(t)\,=\,\mathbf{C}\,x_{i}(t),\end{array}\right.,$$

$$x(t):={\left[x_1(t{)}^T,x_2(t{)}^T,\dots ,x_N(t{)}^T\right]}^T\in {\mathbb{R}}^{Nn}.$$

$$\displaystyle u_{i}(t)=-\sum_{j\in\mathcal{N}_{i}}\mathbf{K}_{ij}\big{(}y_{i}(t)-y_{j}(t)\big{)}$$

$${\nu }_i(t):=z_i(t)-\frac{1}{N}\sum _{j=1}^N{z}_i(t)$$

$$\nu (t)=({\mathbf{M}}_N\otimes {\mathbf{I}}_{m_1})z(t)=({\mathbf{M}}_N\otimes \mathbf{C})x(t),$$

$$u(t)=-(\mathbf{L}\otimes \mathbf{K}\mathbf{H})x(t),$$

$$\rho (\mathbf{L},\mathbf{K}):=\underset{t\to \infty }{\lim }\mathbb{E}\left\{\Vert \nu (t){\Vert }_2^2\right\}.$$

$$\dot{x}=({\mathbf{I}}_N\otimes \mathbf{A}-\mathbf{L}\otimes \mathbf{B}\mathbf{K}\mathbf{H})x+({\mathbf{I}}_n\otimes \mathbf{E})\xi -(\mathbf{L}\otimes \sigma {\mathbf{I}}_{m_3})\eta .$$

$$r:=({\mathbf{U}}^T\otimes {\mathbf{I}}_n)x,\text{leavevmode}\chi :=({\mathbf{U}}^T\otimes {\mathbf{I}}_{m_1})\xi ,\text{leavevmode}\gamma :=({\mathbf{U}}^T\otimes {\mathbf{I}}_{m_3})\eta .$$

$$\displaystyle\Sigma_{i}:\leavevmode\nobreak\ \left.\begin{array}[]{ll}\dot{r}_{i}\,=\,(\mathbf{A}-\lambda_{i}\,{\mathbf{B}}{\mathbf{K}}{\mathbf{H}})\,r_{i}+{\mathbf{E}}\,\chi_{i}-\lambda_{i}{\mathbf{B}}\mathbf{K}\sigma\gamma_{i},\\ \end{array}\right.$$

$$\rho (\mathbf{L},\mathbf{K})=\sum _{i=2}^N\varphi ({\lambda }_i,\mathbf{K}),$$

$$\varphi (\lambda ,\mathbf{K}):=\mathrm{Tr}\left(\mathbf{C}\mathbf{P}(\lambda ,\mathbf{K}){\mathbf{C}}^T\right),$$

$$(\mathbf{A}-\lambda \mathbf{B}\mathbf{KH})\mathbf{P}(\lambda ,\mathbf{K})+\mathbf{P}(\lambda ,\mathbf{K})(\mathbf{A}-\lambda \mathbf{B}\mathbf{K}\mathbf{H}{)}^T$$

$$\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}+\mathbf{E}{\mathbf{E}}^T+{\lambda }^2{\sigma }^2\mathbf{B}\mathbf{K}(\mathbf{B}\mathbf{K}{)}^T=0,$$

$$\varphi (\lambda ,\mathbf{K})={\varphi }_{\xi }(\lambda ,\mathbf{K})+{\sigma }^2{\varphi }_{\eta }(\lambda ,\mathbf{K}),$$

$${\dot{\hat{x}}}_i=\mathbf{A}{\hat{x}}_i+\mathbf{B}{u}_i+{\hat{u}}_i,$$

$$u_i=-\mathbf{K}{\hat{x}}_i.$$

$$\hat{S}_{i}:\left\{\begin{array}[]{l}\begin{bmatrix}\dot{x}_{i}\\ \dot{\hat{x}}_{i}\end{bmatrix}=\begin{bmatrix}\mathbf{A}&-{\mathbf{BK}}\\ \mathbf{0}&\mathbf{A}-{\mathbf{BK}}\end{bmatrix}\begin{bmatrix}x_{i}\\ \hat{x}_{i}\end{bmatrix}+\begin{bmatrix}\mathbf{0}\\ \mathbf{I}_{n}\end{bmatrix}\hat{u}_{i}+\begin{bmatrix}{\mathbf{E}}\\ \mathbf{0}\end{bmatrix}\xi_{i}\vspace{1mm}\\ \hat{y}_{i}=\begin{bmatrix}\mathbf{H}&\mathbf{0}\\ \mathbf{0}&\mathbf{H}\end{bmatrix}\begin{bmatrix}x_{i}\\ \hat{x}_{i}\end{bmatrix}+\begin{bmatrix}{\mathbf{I}_{m_{3}}}\\ \mathbf{0}\end{bmatrix}\eta_{i}\vspace{1mm}\\ z_{i}=\begin{bmatrix}\mathbf{C}&\mathbf{0}\end{bmatrix}\begin{bmatrix}x_{i}\\ \hat{x}_{i}\end{bmatrix}\\ \end{array}\right.,$$

$${\hat{u}}_i=-\hat{\mathbf{F}}\sum _{j\in {\mathcal{N}}_i}{a}_{ij}({\hat{y}}_i-{\hat{y}}_j),$$

$$\displaystyle\hat{\mathbf{F}}=\big{[}-\mathbf{F},\leavevmode\nobreak\ \leavevmode\nobreak\ \mathbf{F}\,\,\,\big{]}\in\mathbb{R}^{n\times(2q)}.$$

$$e(t):=\hat{x}(t)-x(t).$$

$$\delta (t):=({\mathbf{M}}_N\otimes {\mathbf{I}}_n)e(t).$$

$$\mu (\mathbf{L},\mathbf{F}):=\underset{t\to \infty }{\lim }\mathbb{E}\left\{\Vert \delta (t){\Vert }_2^2\right\}.$$

$$\displaystyle\Upsilon_{i}:\leavevmode\nobreak\ \leavevmode\nobreak\ \left.\begin{array}[]{ll}\dot{r}_{i}\,=\,(\mathbf{A}-\lambda_{i}\,{\mathbf{F}}\mathbf{H})\,r_{i}+{\mathbf{E}}\,\chi_{i}-\lambda_{i}\sigma\mathbf{F}\gamma_{i}\\ \end{array}\right.,$$

$$\mu (\mathbf{L},\mathbf{F})=\sum _{i=2}^N\psi ({\lambda }_i,\mathbf{K}),$$

$$\psi (\lambda ,\mathbf{K}):=\mathrm{Tr}\left(\mathbf{Q}(\lambda ,\mathbf{K})\right),$$

$$(\mathbf{A}-\lambda \mathbf{F}\mathbf{H})\mathbf{Q}(\lambda ,\mathbf{K})+\mathbf{Q}(\lambda ,\mathbf{K})(\mathbf{A}-\lambda \mathbf{F}\mathbf{H}{)}^T$$

$$\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}+\mathbf{E}{\mathbf{E}}^T+{\lambda }^2{\sigma }^2{\mathbf{F}}^T\mathbf{F}=0,$$

$${\dot{v}}_i=\mathbf{F}{v}_i+\mathbf{G}{y}_i+\mathbf{TB}{u}_i,$$

$$u_i=\mathbf{K}{\mathbf{Q}}_1\sum _{i=1}^N{a}_{ij}(y_i-y_j)+\mathbf{K}{\mathbf{Q}}_2\sum _{i=1}^N{a}_{ij}(v_i-v_j).$$

Full text

I Introduction

Developing tools to reduce design complexity has been in the center of recent research in networked control systems [1, 2, 3, 4, 5]. In several important applications, network design problem reduces to finding an optimal communication (graph) topology among a network of identical subsystems that are coupled to each other through some common mission-related control objectives. Examples include formation control in a cooperative team of robots, the platoon of vehicles in automated highways, space-time rendezvous in a team of robots, and networks of synchronous oscillators in power networks. The design problem usually involves the optimization of a measure of performance or robustness while respecting various constraints. Due to their combinatorial nature, most network design problems become intractable as network size increases and suffer from high computational complexities. Possibility of characterizing performance and robustness measures in closed and explicit forms will significantly facilitate the design process by allowing the network designer to identify relevant functional properties of the measures and their behaviors with respect to the interconnection topology. In this paper, we present explicit expressions for the $\mathcal{H}_{2}$-norm, as a performance and robustness measure, of a class of interconnected network of linear control systems.

The authors of [6] consider coherency of a platoon of vehicles by evaluating the $\mathcal{H}_{2}$-norm of second-order consensus algorithms and propose several scaling laws for various scenarios of coordination. In [7], ill-posedness of a certain class of platoons is investigated and shown that stabilizability deteriorates as the size of the platoon increases. The string stability of a class of formation problems with limited communication range is studied in [8], where a fundamental limit on the disturbance rejection quality of the network in the frequency domain is derived. The stability and robustness of large platoon of vehicles with double-integrator dynamics are considered in [9], where it is shown that how scaling of a robustness measure (in terms of the platoon size) to external disturbances improves from geometric to polynomial growth when vehicles are allowed to communicate with their two immediate neighbors. In [10], robustness analysis and distributed $\mathcal{H}_{\infty}$ controller design of platoon of vehicles with third-order models and undirected communication topologies are considered. In [11], several graph theoretic bounds on the $\mathcal{H}_{2}$-based performance of linear consensus networks with first- and second-order dynamics are characterized and it is shown how the performance measure scales with the network size and depends on structural properties of the communication topology. In [12], the authors consider distributed $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ controller design for a multi-agent system whose subsystems have general linear time-invariant dynamics. Using a consensus-like algorithm and notion of the grounded graph (e.g., see [13]) to model coupling of agents to leaders, it is shown under what conditions such controllers exist and how they can be suboptimally designed.

In this paper, we consider a network of identical subsystems that are connected over an undirected graph and subject to external disturbance and measurement noise. We propose a methodology to express the steady-state variance of the output of a class of interconnected linear time-invariant networks as a rational function of their Laplacian eigenvalues. Our method extends the existing results in the literature for first- and second-order linear consensus network models (cf. [2] and reference in there). We illustrate that the notion of minimum connectivity threshold is useful for the design of the feedback gains for these networks. It turns out that stabilizability of the nodal dynamics (and detectability in case of observer-based output-feedback) guarantee the existence of such designs. Using these developments, it is shown that fundamental limits may emerge for networks whose subsystems are non-minimum phase. We find graph-theoretic bounds for the performance of the network, which paves the way to find scaling laws for the performance measure. Moreover, a tradeoff between the graph sparsity and performance measure is revealed. Additionally, for networks over path or cycle graphs, we find the asymptotic trend of the performance measure. We bring two extensions of the analyses for the cases of observer-based output feedback as well as a class of composite networks. We have included several parametric and numerical examples to support our theoretical contributions111The proofs are included in the appendices A to P of the paper.. Our approach is advantageous for the design of these dynamical networks. Our spectral expressions can facilitate solving of underlying optimal control problems: instead of dealing directly with optimization problems with high-dimensional matrices, our method leverages the structure of the control system and decouples the roles of typically low-dimensional feedback gains and the eigenspectrum of the communication graph. This is the outgrowth of the preliminary results that were presented in the conference version [14].

II Notations and Preliminaries

The subscripts + and ++ subscripts denote the nonnegative and positive subsets of a set, respectively (e.g. $\mathbb{R}_{+}$). The operator$\mathrm{Tr}(.)$ represents matrix trace. The partial ordering on the cone of positive-semidefinite matrices is denoted via $\succ$ and similar operators. The standard basis for $\mathbb{R}^{N}$ is denoted by the set of vectors $\{e_{1},\dots,e_{N}\}$. The vector and matrix of ones are denoted by ${1}_{N}\in\mathbb{R}^{N}$ and ${\mathbf{J}_{N}}\in\mathbb{R}^{N\times N}$, respectively. Also, $\mathbf{I}_{N}$ and ${\mathbf{M}_{N}}=\mathbf{I}_{N}-\mathbf{J}_{N}/N$ are identity and centering matrices, respectively. The vectorization is denoted by $\mathrm{vec}(\mathbf{S})$. The Kronecker product is denoted by $\mathbf{A}\otimes{\mathbf{B}}$. The matrix transpose and conjugate transpose are denoted by $(.)^{T}$ and $(.)^{*}$ superscripts, respectively. A weighted undirected graph over $N$ nodes is a collection $\mathcal{G}={(\mathcal{V},\mathcal{E},k)}$ with the following components: the set of nodes $\mathcal{V}=\{1,2,\dots,N\}$, the set of edges \mathcal{E}\subset\big{\{}\{i,j\}\;|\;i,j\in\mathcal{V}\}, and the weight function $k:\leavevmode\nobreak\ \mathcal{E}\rightarrow$ $\mathbb{R}_{+}$. We define $k_{ij}:=k(\{i,j\})=k_{ji}$ and form the (symmetric) graph Laplacian ${\mathbf{L}}=\left[l_{ij}\right]\in\mathbb{R}^{N\times N}$ with entries

[TABLE]

The set of neighbors of a node is \mathcal{N}_{i}:=\left\{j\in\mathcal{V}\leavevmode\nobreak\ \big{|}\leavevmode\nobreak\ \{i,j\}\in\mathcal{E}\right\} for $i\in\mathcal{V}$. The eigenvalues of ${\mathbf{L}}$ are denoted by $\lambda_{1}\leq\dots\leq\lambda_{N}$, which are real and nonnegative for a weighted undirected graph. For a connected graph, $\lambda_{1}=0$ with eigenvector $1_{N}$, and $\lambda_{2}>0$. The Laplacian eigendecomposition is $\mathbf{L}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{T},$ where $\mathbf{U}$ is its orthonormal matrix of eigenvectors and $\mathbf{\Lambda}=\mathrm{diag}(\lambda_{1},\dots,\lambda_{N}).$ For positive sequences $\{p_{n}\}_{n\in\mathbb{Z}_{+}}$ and $\{q_{n}\}_{n\in\mathbb{Z}_{+}}$, we adapt $q_{n}=O(p_{n})$ if ${{q_{n}}/{p_{n}}}\leq C\text{ for some }C>0$. Moreover, $q_{n}=O(p_{n})\Leftrightarrow p_{n}=\Omega(q_{n})$. Additionally, $q_{n}=O(p_{n}),\leavevmode\nobreak\ q_{n}=\Omega(p_{n})\Leftrightarrow q_{n}=\Theta(p_{n})$. Finally, we consider $q_{n}\sim p_{n}\Leftrightarrow\displaystyle\lim_{n\rightarrow\infty}{q_{n}}/{p_{n}}=1.$

III Problem Statement

We consider an interconnected network of $N$ subsystems where the dynamics of the $i$’th subsystem is governed by

[TABLE]

for $i=1,\dots,N$, in which $x_{i}(t)\in\mathbb{R}^{n}$ is the state vector of the subsystem, $u_{i}(t)\in\mathbb{R}^{p}$ is the control input, $\xi_{i}(t)\in\mathbb{R}^{m_{1}}$ is the exogenous disturbance input, $\eta_{i}(t)\in\mathbb{R}^{m_{3}}$ is the measurement noise, $y_{i}(t)\in\mathbb{R}^{q}$ is the measurable output, and $z_{i}(t)\in\mathbb{R}^{m_{2}}$ is the performance output. Parameter $\sigma\geq 0$ dictates the magnitude of the measurement noise. The state of the entire network is

[TABLE]

The vectors representing the network input, disturbance, feedback noise, feedback output, and controlled output are similarly defined and denoted by $u$, $\xi$, $\eta$, $y$, and $z$, respectively.

The control objective for the network is to achieve synchronization (or consensus), i.e., $x_{i}(t)-x_{j}(t)\rightarrow 0$ as $t\rightarrow\infty$ for all $i,j\in\{1,\dots,N\}$. To realize this goal, we employ the following feedback control law

[TABLE]

for each subsystem $i\in\{1,\dots,N\}$. The subsystems are allowed to exchange their relative output measurements information over an undirected communication graph $\mathcal{G}$. It is assumed that the structure of the feedback gain matrices $\mathbf{K}_{ij}\in\mathbb{R}^{p\times q}$ are restricted to $\mathbf{K}_{ij}=k_{ij}\mathbf{K}$ where $k_{ij}$’s are nonnegative scalars (i.e., the weights of graph $\mathcal{G}$) and $\mathbf{K}$ is the common factor among all feedback gain matrices.

When stabilizing feedback control law (8) exists and there is no disturbance and noise, one can show that $x_{i}(t)-x_{j}(t)\rightarrow 0$ as $t\rightarrow\infty$ holds for the closed-loop network. However, in the presence of disturbance or noise, the state variables will fluctuate around the consensus state. To quantify these fluctuations, we look at the deviations from the average of the output states subsystems, which are given by

[TABLE]

for every $i\in\mathcal{V}$. We can represent (9) in vector form as

[TABLE]

where ${\mathbf{M}_{N}}$ is the centering matrix of size $N$. The network (7) and (8) asymptotically reaches consensus if and only if $\nu(t)$ asymptotically goes to zero. Since (8) can be rewritten as

[TABLE]

the controller synthesis breaks into two components: designing a feedback gain $\mathbf{K}$ and designing a weighted undirected graph with Laplacian $\mathbf{L}$. It is assumed that measurement noise and noise input are both Gaussian, uncorrelated, and with independent components with unit variance. In order to measure the aggregate fluctuations in the network, we adopt the steady-state variance of the deviation from the average as a measure of performance for the design, which is defined by

[TABLE]

The research problems are to characterize performance measure (12) in terms of Laplacian eigenvalues of the underlying communication graph of the network, illustrate role of feedback (and observer) gains in stability and emergence of fundamental limits on performance and design tradeoffs, and derive scaling laws for the performance as the network grows.

IV Stability and Performance Measure Characterization

We look at the stability criteria for these dynamical networks. Moreover, we derive and characterize spectral expressions for the performance measure. For brevity, we remove the time argument from the variables.

Once we apply feedback control protocol (8), the closed-loop dynamics of the network are given by

[TABLE]

We define the auxiliary variables $r$, $\chi$, and $\gamma$ to be

[TABLE]

Then, the following dynamical decoupling is realized (see [12] for the case of state-feedback without the measurement noise).

Proposition 1**.**

By the change of variables (14), the resulting closed-loop network dynamics given by (13) are decoupled into $N$ systems

[TABLE]

for each $i=1,2,\dots,N$. In the absence of disturbance and noise, the network reaches consensus if and only if systems $\Sigma_{2},\dots,\Sigma_{N}$ are asymptotically stable.

We leverage this decoupling to arrive at spectral expressions for the performance measure of the network.

Theorem 1**.**

Suppose that in (16) systems $\Sigma_{2},\dots,\Sigma_{N}$ are asymptotically stable. Then, the performance measure can be expressed as

[TABLE]

with the performance function $\phi(\lambda,\mathbf{K})$ given by

[TABLE]

which is a rational function of $\lambda$ and entries of $\mathbf{K}$. The map ${\mathbf{P}}(\lambda,\mathbf{K})$ is the unique positive-definite solution to an algebraic Lyapunov equation given by

[TABLE]

for all values of $\lambda$ that make $\mathbf{A}-\lambda{\mathbf{B}}\mathbf{K}\mathbf{H}$ a Hurwitz matrix.

The dimension of the dynamics of each subsystem is often small and has nothing to do with the number of subsystems. Therefore, evaluation of performance function $\phi(\lambda,\mathbf{K})$ can be done via symbolically solving Lyapunov equation (19) after converting it to a linear system by vectorization (see the proof of Theorem 1).

Due to linearity of the Lyapunov equation, one inspects that the performance function $\phi$ can be decomposed into two components according to

[TABLE]

in which spectral functions $\phi_{\xi}$ and $\phi_{\eta}$ only reflect the effect of disturbance and measurement noise, respectively.

Remark 1*.*

In this paper, we occasionally skip argument $\mathbf{K}$ in $\phi(\lambda,\mathbf{K})$ and denote it as $\phi(\lambda)$. In those cases, we solely consider the dependence of the functions on the eigenvalues of the graph Laplacian (i.e., for a fixed feedback gain $\mathbf{K}$).

Remark 2*.*

A part of the result of Theorem 1 is hidden in the analysis provided in [12], in the case of state-feedback. However, the authors did not explicitly derive the spectral expressions for the performance.

We extend the previous analysis to the output-feedback and synthesize a decentralized observer. We show that the separation principal in the linear filtering using Luenberger observers is naturally carried into this design as well. Our procedure consists of four steps:

(i) We augment the dynamics of subsystem $i$ by an observer variable $\hat{x}_{i}\in\mathbb{R}^{n}$, whose dynamics are governed by

[TABLE]

where $\hat{u}_{i}\in\mathbb{R}^{n}$ is an auxiliary control input for the observer. We will set the value of this input in a decentralized manner in the last step.

(ii) As it is usual in the observer design, we use $\hat{x}_{i}$ to compute

[TABLE]

(iii) In addition to the relative output feedback on $\mathbf{H}x_{i}$, the subsystems should share the value of $\mathbf{H}\hat{x}_{i}$ with their neighbors. Once we consider these three steps, the augmented dynamics of subsystem $i$ are

[TABLE]

Variable $\hat{y}_{i}$ has the same role as $y_{i}$ in (7); i.e., the augmented subsystems will use the relative-feedback on this variable.

(iv) We use the following theorem and design the gain of control law (8) when applied on subsystems $\hat{S}_{i}$ in (25), which in this case will be an observer gain.

Theorem 2**.**

Suppose that we apply control law (8) on augmented subsystems $\hat{S}_{i}$ in (25) by setting

[TABLE]

where the observer gain is set to be

[TABLE]

Moreover, assume that $\mathbf{F}\in\mathbb{R}^{n\times q}$ is chosen such that $\mathbf{A}-\lambda_{i}\mathbf{F}\mathbf{H}$ is Hurwitz for $i=2,\dots,N$. Then, the estimation and regulation are separated: if we apply control input $u_{i}$ given in (21) for any $\mathbf{K}$ that makes $\mathbf{A}-{\mathbf{BK}}$ a Hurwitz matrix, then the network with this observer-based relative output-feedback reaches the consensus in the absence of disturbance and noise.

For this design, we denote the performance function by $\phi(\lambda,\mathbf{K},\mathbf{F})$. This function can be found similar to the case of simple state-feedback, except that we need the augmented matrices of $\hat{S}_{i}$ given in (25) for solving (19) and evaluation of this function.

The separation principal together with the duality between the estimation and regulation let us prove similar results for the quality of estimation using this decentralized observer. First, we define the error of estimation as

[TABLE]

Because we are only employing the relative feedback, we may only control the deviations of the error components from their average. These deviations are reflected by the variable

[TABLE]

Next, we define the estimation measure for network as

[TABLE]

The dual of system $\Sigma_{i}$ in (16) is

[TABLE]

which lets us deduce the next result (compare to Theorem 1).

Theorem 3**.**

Suppose that in (32) systems $\Upsilon_{2},\dots,\Upsilon_{N}$ are asymptotically stable. Then, we can express the estimation measure as

[TABLE]

with the estimation function $\psi(\lambda,\mathbf{K})$ given by

[TABLE]

which is a rational function of $\lambda$ and entries of $\mathbf{F}$. The map ${\mathbf{Q}}(\lambda,\mathbf{K})$ is the unique positive-definite solution to an algebraic Lyapunov equation given by

[TABLE]

for all values of $\lambda$ that make $\mathbf{A}-\lambda{\mathbf{F}}\mathbf{H}$ a Hurwitz matrix.

Remark 3*.*

In [15], the authors propose the following observer-based approach. They define their observer variable $v_{i}$ to follow the dynamics

[TABLE]

and define their control law as

[TABLE]

where matrix $\mathbf{F}$ has no eigenvalue in common with $\mathbf{A}$, the pair $(\mathbf{F},\mathbf{G})$ is stabilizable, and $\mathbf{T}$ is the unique solution to Sylvester equation $\mathbf{T}\mathbf{A}-\mathbf{F}\mathbf{T}=\mathbf{G}\mathbf{C}$. Then, they design $\mathbf{K},\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$ such that a design with minimum connectivity threshold is achieved. One can see that our design is different and simpler as we only need a feedback gain $\mathbf{K}$ and an observer gain $\mathbf{F}$. Moreover, our approach is built upon the separation principle between the regulation and estimation, which is also the case in the classical Luenberger (or LQG) observer design.

V Design of Control Law Gains

We investigate the problem of finding feedback gains and focus on gains inducing a minimum connectivity threshold. This property makes the design process with respect to the graph more tractable. After that, we discuss related performance limitations.

V-A * Minimum Connectivity Threshold*

We define the minimum connectivity threshold $\tilde{\lambda}(\mathbf{K})\in[0,\infty]$ for a feedback gain $\mathbf{K}$ to be

[TABLE]

Similar notions have been reported (e.g. [12]), while our goal is characterization of conditions for finding gains with $\tilde{\lambda}(\mathbf{K})<\infty$222$\tilde{\lambda}(\mathbf{K})=\infty$ corresponds to finding the infimum of the empty set in (38).. The following definition is for this purpose.

Definition 1**.**

The feedback gain $\mathbf{K}$ is said to have an unbounded stability region if $\tilde{\lambda}(\mathbf{K})\in[0,\infty)$.

If $\mathbf{K}$ has an unbounded stability region, then the network is robust to all increases in the connectivity: if the network is output-stable for a given graph $\mathcal{G}_{1}$ with Laplacian $\mathbf{L}_{1}$, then for every graph $\mathcal{G}_{2}$ with Laplacian $\mathbf{L}_{2}$ and $\mathcal{G}_{1}\subset\mathcal{G}_{2}$, the network is still output-stable. The reason is that $\lambda_{i}(\mathbf{L}_{1})\leq\lambda_{i}(\mathbf{L}_{2})$ for $i=2,\dots,N$ (this has been emphasized in [12] as well). Moreover, this makes the stability analysis with respect to the graph more tractable, since ensuring $\lambda_{2}(\mathbf{L})>\tilde{\lambda}(\mathbf{K})$ guarantees the output-stability of network. Before bringing methods to find such feedback gains, let us look at a consequence of choosing them.

Theorem 4**.**

For a network designed with a feedback gain $\mathbf{K}$ that in endowed by a connectivity threshold $\tilde{\lambda}(\mathbf{K})<\infty$, the performance function $\phi(\lambda)$ is analytic on interval $(\tilde{\lambda}(\mathbf{K}),\infty)$.

The openness of the interval of interest in Theorem 4 suggests that if $\tilde{\lambda}(\mathbf{K})>0$, we need to maintain a minimum distance from this value. This will make sure that the stability margin is large enough.

V-B *State-Feedback Minimum Connectivity Design *

Let us consider the state-feedback (i.e., $\mathbf{H}=\mathbf{I}_{n}$ in (7)). It turns out that the stabilizability is the necessary and sufficient condition for existence a gain $\mathbf{K}$ that induces a bounded threshold $\tilde{\lambda}(\mathbf{K})$.

Theorem 5**.**

If $(\mathbf{A},{\mathbf{B}})$ is stabilizable, then for every value of $c>0$, the choice of feedback gain given by

[TABLE]

satisfies $\tilde{\lambda}(\mathbf{K})\in[0,c]$, where $\mathbf{Q}\succ 0$ is a solution to the following feasible linear matrix inequality.

[TABLE]

Conversely, if there exists a gain $\mathbf{K}$ with $\tilde{\lambda}(\mathbf{K})<\infty$, then $(\mathbf{A},{\mathbf{B}})$ is stabilizable.

The linear matrix inequality (LMI) (40) is a computational tool to find a gain $\mathbf{K}$ for a given network and graph with a minimum connectivity threshold at most equal to $c$ (see Example 11). The solvability of LMI (40) is called the quadratic stabilizability of $(\mathbf{A},{\mathbf{B}})$ by means of a linear state-feedback (see Section 7.2 of [16]).

Remark 4*.*

This result is inspired by Theorem 11 in [12], while our main contribution is in pointing out the role of stabilizability in existence of feedback gains with minimum connectivity thresholds.

Remark 5*.*

The optimal choice of $\mathbf{Q}$ is not the concern in Theorem 5. Instead, we focus on the existence of designs for $\mathbf{K}$ with a minimum connectivity design. In fact, various performance criteria could potentially get addressed. For instance, suppose that for some $d>0$, we replace LMI (40) with

[TABLE]

Then, for $\mathbf{K}$ computed from (39) using any solution to this inequality $\mathbf{Q}\succ\mathbf{0}$, not only $\tilde{\lambda}(\mathbf{K})\leq c$, but also for each eigenvalue $\lambda>\tilde{\lambda}(\mathbf{K})$, the poles of $\mathbf{A}-\lambda{\mathbf{BK}}$ have real parts less than $-d$ (see [17]). As another example, authors of [12] brought a version of the matrix inequality which ensures that each decoupled subsystem $\Sigma_{i}$ has $\mathcal{H}_{2}$-norm less than a desired value, which they state that could be conservative in practice. Criteria such as robustness or non-fragility could be potentially added by building on top of (40) as well (e.g. see [18]).

V-C Observer-Based Minimum Connectivity Design for Output-Feedback

The duality between the derived conditions on $\mathbf{A}-\lambda_{i}\mathbf{F}\mathbf{H}$ in Theorem 2 and on $\mathbf{A}-\lambda_{i}{\mathbf{B}}{\mathbf{K}}$ in Theorem 5 lets us conclude the following result that resembles the result of Theorem 5.

Theorem 6**.**

Suppose that $(\mathbf{A},\mathbf{H})$ is detectable. Then, for every $c>0$, the following observer gain for the settings of Theorem 2, has an unbounded stability region with $\tilde{\lambda}\left(\mathbf{F}\right)\in\left[0,c\right]$.

[TABLE]

where $\mathbf{Q}\succ 0$ is a solution to the following feasible LMI.

[TABLE]

Conversely, if under the settings of Theorem 4 an observer gain $\mathbf{F}$ has a bounded $\tilde{\lambda}(\mathbf{F})$, then $(\mathbf{A},\mathbf{H})$ is detectable.

The LMI (43) is the quadratic stabilizability condition for the dual pair $(\mathbf{A}^{T},{\mathbf{H}}^{T})$333It is stabilizable since pair $(\mathbf{A},\mathbf{H})$ is detectable.

V-D *Asymptotic Performance and Estimation Bounds *

An important design question is if the performance function $\phi(\lambda,\mathbf{K})$ can be made arbitrarily small, which is related to the notion of almost disturbance decoupling [19]: attenuating the effect of the disturbance in a performance metric as much as desired. We study the case of relative state-feedback below.

Theorem 7**.**

Suppose that $(\mathbf{A},\mathbf{B})$ is stabilizable and $(\mathbf{A},\mathbf{C})$ is detectable and that $\sigma=0$. For all pairs of $\lambda>0$ and $\mathbf{K}$ for which $\mathbf{A}-\lambda{\mathbf{BK}}$ is Hurwitz, the performance function resulting from the relative state-feedback is bounded from below according to

[TABLE]

for a positive semi-definite matrix $\mathbf{P}_{0}$ given by

[TABLE]

where ${\mathbf{P}}_{\epsilon}$ is the unique positive semi-definite solution to the parametric algebraic Riccati equation

[TABLE]

Matrix $\mathbf{P}_{0}$ is zero if and only if transfer matrix $\mathbf{C}(s\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{B}$ is right-invertible and minimum-phase.

For instance, if the transfer matrix is $\mathbf{C}(s\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{B}$ non minimum-phase and the columns of ${\mathbf{E}}$ are not in the null space of $\mathbf{P}_{0}$, then the bound in (44) is strictly positive. The dual of this result for estimation quality is given below, whose proof is identical to Theorem 7.

Theorem 8**.**

Suppose that $(\mathbf{A},\mathbf{E})$ is stabilizable and $(\mathbf{A},\mathbf{H})$ is detectable. If for some gain $\mathbf{F}$, $\mathbf{A}-\lambda_{i}\mathbf{F}\mathbf{H}$ are Hurwitz for $i=2,\dots,N$, then

[TABLE]

for a positive semi-definite matrix $\mathbf{S}_{0}$ given by

[TABLE]

where ${\mathbf{S}}_{\sigma}$ is the unique positive semi-definite solution to the parametric algebraic Riccati equation

[TABLE]

Matrix $\mathbf{S}_{0}$ is zero if and only if transfer matrix $\mathbf{H}(s\mathbf{I}_{n}-\mathbf{A})^{-1}\mathbf{E}$ is right-invertible and minimum-phase.

V-E * Parametric Evaluation of $\tilde{\lambda}(\mathbf{K})$*

In both relative state or output feedback designs, if $n$ is not large (e.g. $n\sim$ 1 to 4), we may design $\mathbf{K}$ with an unbounded stability region using Routh-Hurwitz criteria and explicitly evaluate $\tilde{\lambda}(\mathbf{K})$. In fact, the characteristic equation of the matrix $\mathbf{A}-\lambda{\mathbf{B}}{\mathbf{K}}\mathbf{H}$ for the decoupled systems for eigenvalue $\lambda$ is

[TABLE]

They must be Hurwitz polynomials for $\lambda=\lambda_{2},\dots,\lambda_{N}$. As we enforce the Routh-Hurwitz criteria, we find a set of essentially nonlinear inequalities involving $\lambda$ and elements of $\mathbf{K}$, such that the minimum connectivity threshold is realizable and evaluable based on values of $\mathbf{K}$ (see the next section for examples).

VI Examples of Performance Analysis

In this section, we bring different classes of subsystems and characterize their performance within this framework. We bring additional details of the examples in Appendix Q.

First, we consider two single-input single-output controllable subsystems under the relative state-feedback, where the disturbance and control input drive the dynamics from the same channel (without the measurement noise).

Example 1*.*

Consider the subsystems given in Table I, where we have also reported the corresponding performance functions. For the nodal dynamics $\mathfrak{s}_{1}$ and $\mathfrak{s}_{2}$, supposing that $\mathbf{K}=k>0$ and $\mathbf{K}=[k_{1},k_{2}]\succ 0$, respectively, in both cases $\tilde{\lambda}(\mathbf{K})=0$. Moreover, for $\lambda>\tilde{\lambda}(\mathbf{K})$, performance function $\phi(\lambda)$ is strictly convex and strictly decreasing. If all $a_{i}$’s are zero and $\mathbf{C}=e_{1}^{T}$, these subsystems are called single and double-integrators, respectively. As a numerical example, let us consider double-integrators with $k_{1}=k_{2}=1$. Then, using the second row of Table I we get

[TABLE]

This is a well-known result (e.g. see [20]).

Example 2*.*

Consider double-integrators with relative feedback only on positions (${\mathbf{H}}=[1,0]$) using the decentralized observer of Theorem 6. We let $\mathbf{K}=[k_{1},k_{2}]\succ 0$ and set the observer gain to be $\mathbf{F}=[f_{1},f_{2}]^{T}$. Theorem 2 requires the stability analysis for matrix $\mathbf{A}-\lambda\mathbf{F}{\mathbf{H}}$, which is Hurwitz if and only if $f_{1},f_{2}>0$. Then, we get $\tilde{\lambda}(\mathbf{F})=0$. We can show that

[TABLE]

where $c_{1}$ to $c_{8}$ are polynomials of $k_{1},k_{2},f_{1},$ and $f_{2}$. Using the observer with $k_{1}=k_{2}=f_{1}=f_{2}=1$, (51) becomes

[TABLE]

One observes that for weak connectivity regimes (i.e., $\lambda$ near zero), $\phi(\lambda)$ in (52) is close to the function in (50), while as $\lambda$ increases, the performance function corresponding to relative state-feedback vanishes, while the function from observer design does not.

Example 3*.*

We consider a triple-integrator with dynamics

[TABLE]

Let us choose the state to be $[x_{i},\dot{x}_{i},\ddot{x}_{i}]^{T}$ with element-wise positive gain $\mathbf{K}=[k_{1},k_{2},k_{3}]\succ 0$. We can show that

[TABLE]

The next two examples also have performance functions that under conditions become strictly decreasing and convex.

Example 4*.*

The dynamics of a harmonic oscillator of mass $m$ are governed by

[TABLE]

where $\zeta$ is the damping ratio and $\omega_{0}$ is the undamped angular frequency (see [21]). We consider ${\mathbf{C}}=[1,0]$ and compute $\phi(\lambda)$ with the relative state-feedback on $[x,\dot{x}]^{T}$ with $\mathbf{K}=[k_{1},k_{2}]\succ 0$. Using arguments similar to Example 1, if we define $\alpha_{1}:={m\omega_{0}^{2}}/{k_{1}}$ and $\alpha_{2}:={2m\omega_{0}\zeta}/{k_{2}}$ we get

[TABLE]

Again, for element-wise positive feedback gains, $\phi(\lambda)$ is strictly convex and strictly decreasing for $\lambda>\tilde{\lambda}(\mathbf{K})=0$.

Example 5* (Platoon of Vehicles).*

We consider a network of vehicles, in which the position of $i$’th vehicle is denoted by $p_{i}\in\mathbb{R}$. It has the third-order dynamics

[TABLE]

where the input $u_{i}\in\mathbb{R}$ is the desired acceleration and $\xi_{i}\in\mathbb{R}$ is the disturbance. The time-constant $\tau>0$ characterizes how fast the vehicles responds to the acceleration command. The state vector is chosen as $[p_{i},\dot{p}_{i},\ddot{p}_{i}]^{T}$, where they denote the (errors in) the position, velocity, and acceleration of the vehicles in the platoon, respectively (see [10] for more details). The state-space matrices are given in the appendix. Using relative state-feedback, by application of the Routh-Hurwitz criteria we find that if $\mathbf{K}=[k_{1},k_{2},k_{3}]$ satisfies $k_{1},k_{2}>0,k_{3}\geq 0,$ we get

[TABLE]

We can show that if $\sigma=0$, we get

[TABLE]

If $k_{3}=0$, the design corresponds to a relative output-feedback on only positions and velocities, with a performance function

[TABLE]

which is strictly convex and strictly decreasing for $\lambda>\tilde{\lambda}(\mathbf{K})=0$. If $k_{3}>0$, we have the relative state-feedback and for $\lambda>\tilde{\lambda}(\mathbf{K})$ the same argument holds (see the appendix).

Example 6*.*

Consider a network with nodal matrices

[TABLE]

for $\zeta>0$ and $\sigma=0$. We observer that the subsystems have a non minimum-phase input-output transfer function

[TABLE]

where $\zeta>0$ is the location of the right-hand plane zero. Let us consider the relative state-feedback. We can show that in this case, we have $\mathbf{P}_{0}=\mathrm{diag}\left(2\zeta,0\right)$. For a disturbance matrix ${\mathbf{E}}=\left[\alpha,\beta\right]^{T}$, Theorem 7 gives us the bound

[TABLE]

Alternatively, if we use the relative state-feedback, we can show that the corresponding performance function ${\phi}(\lambda,\mathbf{K})$ is

[TABLE]

which is strictly convex and decreasing for $\lambda>\tilde{\lambda}(\mathbf{K})=0$. Now, for any gain $\mathbf{K}$ with an unbounded stability region

[TABLE]

where $r:={k_{2}}/{k_{1}}$. By differentiation with respect to $r$, we find that the right side attains its minimum at $r={1}/{\zeta}$. Thus

[TABLE]

which is the same bound as (62). The bound on the performance function scales with the magnitude of the right-hand plane zero at $\zeta$. One inspects that if the disturbance enters the subsystem from the same channel as the control input, we do not face a fundamental limitation on the performance, because in this case it does not touch the zero dynamics of the subsystems (see [22] for a similar observation in the case of $\mathcal{H}_{\infty}$-norm).

Example 7*.*

In this example, first, we consider two different designs for a network of double-integrator agents with measurement noise. Recall that the magnitude of feedback noises is controlled by parameter $\sigma>0$.

(i) the relative state-feedback without the filtering (i.e., without the decentralized observer): in this case, using $\mathbf{K}=[k_{1},k_{2}]$, we can show that

[TABLE]

in which the first term can be recovered from Table I and the second term appears due to the measurement noises.

(ii) the relative output-feedback on positions with the decentralized observer: in this case

[TABLE]

in which $\phi_{\xi}$ is the performance function read from (51) and $c_{9}$ to $c_{14}$ are polynomials of $k_{i}$’s and $f_{i}$’s. For instance, in the case of $k_{1}=k_{2}=f_{1}=f_{2}=1$, this function becomes

[TABLE]

Next, we find estimation function $\psi(\lambda,\mathbf{F})$. We can show that

[TABLE]

The first term is due to the disturbances, while the second term originates from the feedback noises. The transfer matrix $\mathbf{H}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{E}$ is right-invertible and minimum-phase. Hence, as Theorem 8 suggests, as $\sigma$ becomes smaller, we can make the estimation arbitrarily precise by increasing the magnitude of observer gains $f_{1}$ and $f_{2}$.

VII Analysis of Network of Networks

We introduce and analyze a class of networks of networks that are built by a repeated application of control law (8). For simplicity of the developments, we neglect the feedback noises (i.e., set $\sigma=0$). One can show that the same approach works in the presence of those noises as well.

VII-A Construction Procedure for Composite Networks

First, we build identical networks using control law (8) over graph $\mathcal{G}_{1}$. We denote the number of nodes of $\mathcal{G}_{1}$ by $m$ and the order of the state-space realization for each subsystem by $n$. Moreover, we denote the feedback gain used to build each network by $\mathbf{K}_{1}\in\mathbb{R}^{p\times q}$. Let us denote the state of subsystem $j$ in module or subnetwork $i$ by $x_{j}^{(i)}\in\mathbb{R}^{n}$. Similarly, we denote the rest of corresponding variables. The Laplacian matrix corresponding to $\mathcal{G}_{1}$ is also denoted by $\mathbf{L}_{1}$. For the subsequent analysis, let us define

[TABLE]

According to (13), the dynamics of the subnetwork $i$ are

[TABLE]

where $x^{(i)}:=\begin{bmatrix}(x_{1}^{(i)})^{T}\,\dots\,(x_{m}^{(i)})^{T}\end{bmatrix}^{T}\in\mathbb{R}^{mn},$ is the state vector of module $i$ and disturbance vector $\xi^{(i)}\in\mathbb{R}^{mm_{1}}$ is defined similarly. Without loss of generality, we designate the last node in graph $\mathcal{G}_{1}$ as the port of the module444 If we wish to choose another node, we can simply relabel the nodes., which corresponds to a subsystem that we can add a term to its control input. This converts (70) to new open-loop dynamics

[TABLE]

wherein $u^{(i)}\in\mathbb{R}^{m}$ is the tunable control input to the module. Moreover, we assume that two modules can become interconnected only through their port nodes. Then, the only variable that module $i$ can use for relative feedback is the output variable for the port node, which is denoted by

[TABLE]

We collect $N$ instances of these networks with dynamics (71) and feedback variables (72) to construct a composite network. Therefore, the subsystems equivalent to $S_{i}$ in (7) for this network design are

[TABLE]

with the structured matrices defined by (69). Now, we build a modular network by application of control law (8) with $N$ modules (or subnetworks) connected over a higher level graph $\mathcal{G}_{2}$ with feedback $\mathbf{K}_{2}\in\mathbb{R}^{p\times q}$. 555 Feedback gains $\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ are matrices of the same dimension because we have chosen one node as the port of a subnetwork. If $\{i,j\}\in\mathcal{E}_{2}$ has a weight denoted by $b_{ij}$, then the application of control law (8) will be

[TABLE]

We have $N$ modules and each one consists of $m$ subsystems. Therefore, the consensus output of the network should be

[TABLE]

Then, we set its steady-state variance as the performance measure

[TABLE]

where $\mathbf{L}_{2}$ is the graph Laplacian of $\mathcal{G}_{2}$. In Fig. 1, we illustrate this composite structure using an example: we have four modules and inside each of them, three subsystems are interconnected over graph $\mathcal{G}_{1}$, in this case a complete graph. These subnetworks are then connected via their ports over graph $\mathcal{G}_{2}$, which in this case is a path.

*Interpretation of Construction: * Let say modules $i$ and $j$ are connected, thus $\{i,j\}\in\mathcal{E}_{2}$. Then, the ports of these two modules will have access to the relative difference of their feedback output $y_{m}^{(i)}-y_{m}^{(j)}$ and will reflect this feedback term in their control input. Mathematically speaking, the input to the port node666 Recall that the port node is arbitrary chosen or labeled to be number $m$. in module $i$ is

[TABLE]

The first term is due to initial application of control law (8) over $\mathcal{G}_{1}$ with an edge set $\mathcal{E}_{1}$, while the second term is $u^{(i)}$ from (77) based on the composite network design over $\mathcal{G}_{2}$.

VII-B Stability and Performance of Composite Networks

Theorem 9**.**

Consider a dynamical network over graph $\mathcal{G}_{1}$ with a bounded performance measure $\rho(\mathbf{L}_{1},\mathbf{K}_{1})$. Suppose that in Proposition 1 and Theorem 1 we apply control law (8) on systems $S^{(i)}$ defined in (76) over $\mathcal{G}_{2}$ with feedback gain $\mathbf{K}_{2}$. The resulting composite network reaches consensus if and only if $\left.\tilde{\mathbf{A}}-\lambda_{i}(\mathbf{L}_{2})\tilde{\mathbf{B}}{\mathbf{K}}_{2}\tilde{\mathbf{H}}\right.$ is Hurwitz for nonzero eigenvalues of $\mathbf{L}_{2}$. Moreover, if $\phi_{\mathrm{nn}}(\lambda,\mathbf{K})$ is the performance function derived from Theorem 1 for subsystems $S^{(i)}$ defined in (76), then

[TABLE]

The significance of this result is that for a fixed module graph $\mathcal{G}_{1}$ with Laplacian $\mathbf{L}_{1}$ and $\mathbf{K}_{1}$, the value of $\rho(\mathbf{L}_{1},\mathbf{K}_{1})$ and the form of composite performance function $\phi_{\mathrm{nn}}$ are fixed. Thus, we can quantify the role of higher level graph $\mathcal{G}_{2}$ and feedback gain $\mathbf{K}_{2}$ in the performance of the composite network by looking at the second term. The extra term compared to Theorem 1 appears because

[TABLE]

where the right-hand side is the output that would have resulted in an expression of form (17).

Remark 6*.*

If a subnetwork is one subsystem, then (81) reduces to (17), since each subsystem as a network satisfies $\rho(\mathbf{L}_{1},\mathbf{K}_{1})=0$.

VII-C * Minimum Connectivity Design For Composite Networks*

We show that if $\tilde{\lambda}(\mathbf{K}_{1})<\infty$, then there exists a simple choice for $\mathbf{K}_{2}$ such that it has also an unbounded stability region in terms of the eigenvalues of higher level Laplacian $\mathbf{L}_{2}$; i.e., $\tilde{\lambda}(\mathbf{K}_{2})$ exists and if $\lambda>\tilde{\lambda}(\mathbf{K}_{2})$ then $\left.\tilde{\mathbf{A}}-\lambda\tilde{\mathbf{B}}{\mathbf{K}}_{2}\tilde{\mathbf{H}}\right.$ is Hurwitz. This would remedy the concerns about possible complexities in the design of $\mathbf{K}_{2}$ over graph $\mathcal{G}_{2}$ .

Theorem 10**.**

Suppose that the subnetworks are built over any graph $\mathcal{G}_{1}$ and feedback gain $\mathbf{K}_{1}$, which has an unbounded stability region. For any $\alpha>0$, let us choose the feedback gain of the composite network to be $\mathbf{K}_{2}=\alpha\mathbf{K}_{1}$. Then, $\mathbf{K}_{2}$ has an unbounded stability region with respect to the eigenvalues of higher level Laplacian $\mathbf{L}_{2}$.

This result is simplified if $\tilde{\lambda}(\mathbf{K}_{1})=0$.

Corollary 1**.**

Suppose that the subnetworks of the network of networks are built with $\mathbf{K}_{1}$, which induces $\tilde{\lambda}(\mathbf{K}_{1})=0$. Let us choose $\mathbf{K}_{2}=\alpha\mathbf{K}_{1}$ for some $\alpha>0$ in the design of the described composite networks. Then, higher level feedback gain $\mathbf{K}_{2}$ satisfies $\tilde{\lambda}(\mathbf{K}_{2})=0$ with respect to the eigenvalues of higher level Laplacian $\mathbf{L}_{2}$.

VII-D * Examples of Networks of Networks*

Example 8*.*

Consider a modular network with subnetworks of single-integrators over an unweighted path graph $\mathcal{G}_{1}$ of $m$ nodes, where the last node of the module is its port. We choose $\mathbf{K}_{1}=k_{1}>0,$ so the open-loop dynamics of the modules before design of the composite network based on (71) are

[TABLE]

where $x^{(i)}\in\mathbb{R}^{m},\xi^{(i)}\in\mathbb{R}^{m}$, $u^{(i)}\in\mathbb{R}$, and $\mathbf{L}_{1}$ is Laplacian of the unweighted path graph over $m$ nodes. Then, choosing $\mathbf{K}_{2}=k_{2}>0$, the performance function of the composite network is

[TABLE]

Based on Corollary 1 in the higher level $\tilde{\lambda}(\mathbf{K}_{2})=0$. Moreover, we inspect that the resulting family of functions is strictly convex and decreasing for $\lambda>0$ (see the appendix for the details).

Example 9*.*

Consider a modular network, where each subnetwork consists of subsystems with the single or double-integrator dynamics. In this case, we set $\mathcal{G}_{1}$ to be the unweighted complete graph over $m$ nodes (similar to the example illustrated in Fig. 1 for the subnetworks with $m=3$). Therefore, unlike Example 8, no matter which node is chosen as the port, the subnetwork will be the identical. For element-wise positive feedback gains $\mathbf{K}_{1}$ and $\mathbf{K}_{2}$, Corollary 1 again implies that the minimum connectivity threshold in terms of the eigenvalues of $\mathbf{L}_{2}$ for both nodal dynamics is zero. Moreover, let $\mathbf{K}_{1}=\mathbf{K}_{2}=k>0$ for the single-integrators and $\mathbf{K}_{1}=\mathbf{K}_{2}=[k_{1},k_{2}]\succ 0$ for the double-integrators (for simplicity). Using the solution to the Lyapunov equation, we find performance function $\phi_{\mathrm{nn}}(\lambda)$ for these subnetworks, which are e given in Table II. These functions are strictly convex and decreasing for $\lambda>0$. As a sanity check, for $m=2$ the unweighted path and complete graphs coincide and the first formula in Table II and (84) produce identical functions for $k_{1}=k_{2}=k$ (see appendix Q for details).

VIII Performance Bounds and Scaling

We look at the cases where combining the information on graph parameters and derived performance functions can give us macroscopic information on the performance measure.

VIII-A * Performance Bounds and Scaling *

In Sections VI and VII, a majority of the derived performance functions are convex. In what follows, we show that this property is useful in derivation of performance bounds.

Theorem 11**.**

Consider a network of $N$ subsystems with a performance functions $\phi(\lambda)$ that is convex. The performance measure over an unweighted graph with $M$ edges and maximum nodal degree of $\Delta$ is lower-bounded according to

[TABLE]

where the equality holds if and only if graph $\mathcal{G}$ is either complete graph or star graph.

Example 10*.*

Consider a network with nodal dynamics $\mathfrak{s}_{2}$, $a_{1}=0$ and $b_{0}=b_{1}=k_{1}=k_{2}=a_{2}=1$. For all connected unweighted graphs with $3$ to $7$ nodes, we do a survey for the ratio of the sides of inequality (85), that is

[TABLE]

The distribution of $\mathfrak{r}_{1}$ versus $N$ is illustrated in Fig. 2. As $N$ increases, it tends to an almost fixed curve (with a growing tail), where $\sim 90\%$ of the graphs induce a ratio $\mathfrak{r}_{1}<2$.

Theorem 12**.**

Consider a network of $N$ subsystems with a performance functions $\phi(\lambda)$ that is convex. The performance measure over any weighted graph with a total weight of $W$ is lower-bounded according to

[TABLE]

*where the equality holds if and only if the graph is complete and with identical weights. *

Theorems 11 and 12 give rules of thumb about the best achievable performance. To do so, we combine information on the nodal dynamics (through the form of the performance function) and macroscopic graph information.

Corollary 2**.**

Under the settings of Theorem 12, it holds that

[TABLE]

Performance-Sparsity Tradeoff: Suppose that $\phi(\lambda)$ is also decreasing. For an unweighted graph, $W=M$. Therefore, we can reorganize the result of Theorem 12 and write

[TABLE]

This result is useful in quantification of the following tradeoff: as the graph of the network becomes sparser, the best attainable value of the performance measure will increase. The following example highlights two specific cases.

Example 1 (Continued)**.

For networks with subsystems that have $\mathfrak{s}_{1}$ dynamics, Corollary 2 implies that over any unweighted graph

[TABLE]

For networks with subsystems of $\mathfrak{s}_{2}$ dynamics we deduce

[TABLE]

For instance, in the special case of $a_{1}=a_{2}=0$ for $\mathfrak{s}_{2}$ we get

[TABLE]

that clearly reflects the sparsity-performance tradeoff for a consensus network of double-integrators (see [11] for a similar result for single-integrator agents).

VIII-B * Performance Asymptotic over Path and Cycles*

Theorem 13**.**

For a network of $N$ subsystems over an unweighted path or cycle graph with $\tilde{\lambda}(\mathbf{K})=0$, it holds that

[TABLE]

where $\Gamma_{N}$ can be computed using a parametric integral

[TABLE]

Moreover, if $\phi(\lambda)$ is bounded at $\lambda=0$, then it holds that

[TABLE]

Corollary 3**.**

The performance measure scales similarly with respect to $N$ over unweighted path and cycle graphs. Moreover, if $\phi(\lambda)$ is bounded at [math], the performance measure over the paths and cycles converge to the same value as $N\rightarrow\infty$.

We should emphasize on few points: (i) Theorem 13 does not depend on neither convexity nor monotonicity of $\phi(\lambda)$; (ii) the requirement $\tilde{\lambda}(\mathbf{K})=0$ is natural, since as $N$ increases $\lambda_{2}(\mathbf{L})=\Theta(1/N^{2})$; i.e., it becomes arbitrary small. Otherwise, there exist $N_{1}$ such that for $N\geq N_{1}$, $\lambda_{2}<\tilde{\lambda}(\mathbf{K})$. (iii) This approximation idea has been previously reported, e.g. in [23] it is used for estimation of Estrada index. However, we find the reason for which the approximations find the scaling of the sums, even if $\phi(\lambda)$ is singular at $\lambda=0$.

Example 1 (Continued)**.

We apply Theorem 13 on a network of $\mathfrak{s}_{1}$ subsystems with $a=0$ (i.e., single-integrators) over an unweighted path and arrive at the asymptotic expression

[TABLE]

while if $a>0$, for $\alpha:=a/k$, we have the approximation

[TABLE]

For $\mathfrak{s}_{2}$ agents with $a_{0}=a_{1}=0$, the performance measure satisfies

[TABLE]

Next, for these agents with $b_{1}=b_{0}=k_{1}=k_{2}=1$ over an unweighted path graph of $N=10,15,\dots,100$ nodes, we investigate the claim of Theorem 13 by looking at the ratio

[TABLE]

with $h$ given in (99). The result is shown in Fig. 3, where according to Theorem 13, $\mathfrak{r}_{2}$ indeed goes to a constant.

Example 4 (Continued)**.

For a network of harmonic oscillators with $\alpha_{1}\neq\alpha_{2}$ over a path graph, Theorem 13 implies that

[TABLE]

We call the right hand side $f(N,\alpha_{1},\alpha_{2})$. To empirically examine the gap, we consider

[TABLE]

We set $k_{1}=k_{2}=1$, $\alpha_{1}=2\alpha_{2}$ for $\alpha_{2}\in[0.4,4]$, and vary $N$ between $10$ and $200$. Because $\phi(\lambda)$ is bounded, as $N$ increases the approximation becomes tighter as shown in Fig. 4.

Example 5 (Continued)* (Platoon over a Path).*

For a platoon of vehicles over a path graph, Theorem 13 suggests that $\rho$ scales with $N^{4}$. Thus, the $\mathcal{H}_{2}$-norm scales with $N^{2}$. As reported by the authors in [10], a similar scaling law in the case of $\mathcal{H}_{\infty}$ norm of the network over this topology holds (with an additional leader).

Example 7 (Continued)**.

We can apply Theorem 13 to find the scaling for the estimation measure as well. Similar to (99), we can show that the estimation measure in a network of double-integrators over a path graph satisfies

[TABLE]

Example 8 (Continued)**.

We consider a network of $N$ subnetworks over a path graph, where the subsystems are a network of single integrators, also over a path graph with $m$ subsystems as analyzed in Example 8. This network is illustrated in Fig. 5. From (97), we already know that the performance of isolated subnetworks satisfies

[TABLE]

Combining (103) and Theorem 7, we can show that

[TABLE]

IX Application to Formation of Aircraft

Example 11* (Formation of Aircraft).*

We consider a linearized model for the dynamics of an aircraft [24] expressed as

[TABLE]

where $X=\begin{bmatrix}\mathrm{u}\leavevmode\nobreak\ \mathrm{v}\leavevmode\nobreak\ \dot{\theta}\leavevmode\nobreak\ \theta\leavevmode\nobreak\ \mathrm{x}\leavevmode\nobreak\ \mathrm{z}\end{bmatrix}^{T}$ (see Appendix Q for the numerics). The variable $\mathrm{u}$ is the horizontal velocity component from its set point and $\mathrm{v}$ is the component normal to that. The pitch angle is denoted by $\theta$. The control inputs $u_{1}$ and $u_{2}$ are the elevator angle and thrust force, respectively. The scalars $\xi_{1}$ and $\xi_{2}$ denote the wind velocity in the longitudinal and lateral directions, respectively. We consider the formation shown in Fig. 6. Once each vehicle takes into account the relative distances from their neighbors (in computation of the position feedbacks), we can use these dynamics to analyze the performance of this network with the performance output $z=\begin{bmatrix}\alpha\mathrm{x}\,\,\,\beta\mathrm{z}\end{bmatrix}^{T}.$

*Relative State-Feedback: * We use the convex optimization toolbox CVX [25] to find $\mathbf{K}$ for $c=0.25$ using Theorem 5. If $\xi_{1}$ and $\xi_{2}$ have intensity of unity, we get

[TABLE]

where $\phi_{1}(\lambda)$ and $\phi_{2}(\lambda)$ describe the magnitude of the fluctuations in the formation in $\mathrm{x}$ and $\mathrm{z}$ directions, respectively. The performance functions are rational functions with the numerator and denominator of order 9. While it is guaranteed to get $\tilde{\lambda}(\mathbf{K})\in[0,0.25]$, we have $\tilde{\lambda}(\mathbf{K})=0$. In Fig. 7, we plot $\phi_{1}$ and $\phi_{2}$, where for larger values of $\lambda$, they are different by more than an order of magnitude. The function $\phi_{2}(\lambda)$ is convex and decreasing, while $\phi_{1}(\lambda)$ is neither strictly convex nor monotone for $\lambda>0$. This suggests that properties of these functions in general could be beyond a simple classification.

*Observer-Based Relative Output-Feedback: * next, we consider the observer-based output-feedback on the last two states of each subsystem (i.e., horizontal and vertical relative positions). For the value of $\mathbf{K}$ we reuse its value from the previous design. We choose observer gain $\mathbf{F}$ for $c=0.25$ using Theorem 6 and find that

[TABLE]

where these two functions are also depicted in Fig. 7. In this case, $\tilde{\lambda}(\mathbf{F})=0$ as well. In Fig. 8, we demonstrate two sample longitudinal output plots based on these two designs, where we have $5$ planes that are supposed to travel with $\Delta\mathrm{x}=0.6$. The graph is a path with weights of $4$ and identical disturbance samples are fed into the subsystems in two cases. The different level of fluctuations is justifiable upon comparison of the values of $\phi_{1}$ and $\hat{\phi}_{1}$ in Fig. 7.

X Conclusion and Discussion

We brought a unifying framework for performance analysis of a class of networked control systems. The resulting spectral expressions let us derive bounds and scaling laws for the performance of the system. We would like to include a a number final remarks:

(i) The spectral expressions for the performance measure can be used to find the optimal values of feedback gain $\mathbf{K}$. In fact, for large networks, solving the Lyapunov equation for the $\mathcal{H}_{2}$ performance measure once the value of feedback gain is updated could be computationally expensive. Instead, suppose that we find the spectral expressions for the performance measure for a fixed graph. Then, our objective function will be a scalar function of the feedback gain. The resulting problem can be effectively approached using general nonlinear problem methods. This approach is also useful when solving for optimal observer gains $\mathbf{F}$ or feedback gains for composite networks $\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ when the graph is fixed. The gains derived from the linear matrix inequalities given in Section V can be used as a starting point of the optimization procedure.

(ii) We can derive similar spectral expressions for the variance of the control input that is consumed throughout the network in the steady-state, which is given by

[TABLE]

Then, we can show that

[TABLE]

for rational input function $\phi_{u}(\lambda,\mathbf{K})$ given by

[TABLE]

The map ${\mathbf{P}}(\lambda,\mathbf{K})\succ 0$ is the solution to (19). For instance, we can show the input functions for networks of single-integrators and double-integrators are given by

[TABLE]

respectively. The developments in this paper which has to do with the performance functions can be applied to the input functions as well (e.g. asymptotic control input over a path).

(iii) In cases that symbolic evaluation of the performance functions is computationally prohibitive, an alternative option is to conduct regression to estimate the coefficients of these rational performance functions numerically.

Appendix A

Proof of Theorem 1: Let us define $m=m_{1}+m_{3}$ The transfer matrix from disturbance and noise $[\xi^{T},\eta^{T}]^{T}$ to consensus output $\nu$ can be expressed as

[TABLE]

where $\tilde{\mathbf{G}}_{i}(s)$ is the transfer matrix from $\left[\chi_{i}^{T},\gamma_{i}^{T}\right]^{T}$ to $r_{i}$. This lets us compute the following quantity

[TABLE]

The matrix $\mathbf{U}^{T}{\mathbf{M}_{N}}{\mathbf{M}_{N}}\mathbf{U}$ is simply given by

[TABLE]

Taking ${1}/(2\pi)\int_{-\infty}^{\infty}\mathrm{Tr}(.)\leavevmode\nobreak\ d\omega$ from the both sides results in

[TABLE]

Due to cyclic property of the trace, the first and last matrix in the trace argument cancel out and we get

[TABLE]

The last $\mathcal{H}_{2}$ norm term can be computed using the state-space formulation of systems $\Sigma_{i}$. This will be the Lyapunov equation (19) [26]. Therefore, we have managed to prove

[TABLE]

Next, we prove that $\phi(\lambda,\mathbf{K})$ is a rational function. The Lyapunov equation (19) upon vectorization becomes

[TABLE]

Using the Cramer’s rule, for $j=1,2,\dots,n^{2}$, we may compute the $j$’th element of $\mathrm{vec}(\mathbf{P})$ as

[TABLE]

where $(\mathbf{D})_{-j}$ is the matrix derived by replacing column $j$ of $\mathbf{D}$ with $-\mathrm{vec}({\mathbf{E}}{\mathbf{E}}^{T}+\lambda^{2}\sigma^{2}{\mathbf{B}}{\mathbf{K}}({\mathbf{B}}{\mathbf{K}})^{T})$. Both numerator and denominator are polynomials of $\lambda$ with coefficients that are polynomials of the elements of $\mathbf{K}$. Therefore, the same conclusions holds about $\phi(\lambda,\mathbf{K})=\mathrm{Tr}(\mathbf{C}{\mathbf{P}}(\lambda,\mathbf{K})\mathbf{C}^{T})$.

Appendix B

Proof of Theorem 2: If we apply control law (8) for the new subsystem $\hat{S}_{i}$ with observer gain $\hat{\mathbf{F}}=[-\mathbf{F},\mathbf{F}]$ (partitioned based on $\hat{y}_{i}$), we have the following formula

[TABLE]

This means that $\Sigma_{i}$ corresponding to the dynamics are

[TABLE]

Defining the error as $e_{i}:=r_{i}-\hat{r}_{i}$, we get that

[TABLE]

The subsystems in the consensus problem have reduced to the familiar decoupled Leunberger observer/regulator form (e.g. see [27]). Therefore, we need to simultaneously have: ${\mathbf{A}}-\lambda_{i}\mathbf{F}\mathbf{H}$ for $i=2,\dots,N$ and ${\mathbf{A}}-{\mathbf{BK}}$ to be Hurwitz. Then, the corresponding $\Sigma_{i}$ is asymptotically stable for $i=2,\dots,N$ and the network reaches the consensus.

Appendix C

Proof of Theorem 3: If we consider the dynamics of $e_{i}$ in (110), they are identical to dynamics of $\Upsilon_{i}$ in (32). The rest of the proof is similar to Theorem 1 once we replace $\Sigma_{i}$ with $\Upsilon_{i}$.

Appendix D

Proof of Theorem 4: The definition of $\tilde{\lambda}(\mathbf{K})$ implies that for all $\lambda>\tilde{\lambda}(\mathbf{K})$, subsystems $\Sigma_{2}$ to $\Sigma_{N}$ are asymptotically stable. Therefore, the performance function is bounded. Because $\phi(\lambda)$ is rational, it is analytic everywhere in its domain, including this interval.

Appendix E

Proof of Theorem 5: First, for a linear time invariant control system the feasibility of the linear matrix inequality and the stabilizability are equivalent [16]. The second part of the claim is a special case of Theorem 11 in [12] with only accounting for the stabilizablity of the subsystems, so we do not repeat the proof in this manuscript. The converse argument holds because if $\tilde{\lambda}(\mathbf{K})<\infty$, then for $\mathbf{K}^{*}=2\tilde{\lambda}(\mathbf{K})\mathbf{K}$, $\mathbf{A}-\mathbf{BK}^{*}$ is Hurwitz; i.e., $(\mathbf{A},\mathbf{B})$ is stabilizable.

Appendix F

Proof of Theorem 6: Due to duality of between the stabilizability and detectability, if the pair $({\mathbf{A}},{\mathbf{H}})$ is detectable, then $({\mathbf{A}}^{T},{\mathbf{H}}^{T})$ is stabilizable. Now, we can use the same argument as Theorem 5 to complete the proof.

Appendix G

Proof of Theorem 7: Consider the control system

[TABLE]

Saberi et. al. [28] have shown that the minimum value of the $\mathcal{H}_{2}$ norm for this system is $\gamma^{*}(\epsilon)=\sqrt{\mathrm{Tr}\left({\mathbf{E}}{\mathbf{P}}_{\epsilon}{\mathbf{E}}^{T}\right)},$ where ${\mathbf{P}}_{\epsilon}$ can be computed as the solution to an algebraic Riccati equation

[TABLE]

Moreover, it has been shown that if $({\mathbf{A}},\mathbf{B})$ is stabilizable and $({\mathbf{A}},\mathbf{C})$ is detectable, then ${\mathbf{P}}_{\epsilon}$ converges to zero if and only if the mentioned transfer matrix is right-invertible and minimum-phase. Now, we should note that the performance function is the $\mathcal{H}_{2}$ norm squared of a system similar to (113), expect that we have $\mathbf{B}\rightarrow\lambda\mathbf{B}$. Under this modification, the limiting case for ${\mathbf{P}}_{\epsilon}$ in Riccati equation (114) does not change.

Appendix H

Proof of Theorem 9: Let us denote orthonormal eigendecompostion of Laplacian $\mathbf{L}_{2}$ by $\mathbf{L}_{2}=\mathbf{U}_{2}\mathbf{\Lambda}_{2}\mathbf{U}_{2}^{T}$. The decoupled system $\Sigma_{i}$ in this case is

[TABLE]

Let us call the transfer matrix from $\chi^{(i)}$ to ${r}^{(i)}$ by $\tilde{\mathbf{G}}^{(i)}$. Then, the transfer matrix from disturbance $\xi$ to performance output $\nu_{\mathrm{nn}}$ in the case of network of networks can be written as

[TABLE]

This lets us compute the following quantity

[TABLE]

We take the trace and move the first two terms of the trace argument to the right to get

[TABLE]

The intermediate term can be simplified according to

[TABLE]

Hence, we can further write

[TABLE]

If we take the map ${1}/(2\pi)\int_{-\infty}^{\infty}\mathrm{Tr}(.)\leavevmode\nobreak\ d\omega$ from the sides

[TABLE]

For $i=1$, $\lambda_{1}(\mathbf{L}_{2})=0$ and the system $\Sigma_{1}$ will have a transfer matrix from the disturbance to output $(\mathbf{M}_{m}\otimes\mathbf{C})\tilde{\mathbf{G}}^{(1)}$. Moreover, in this case $\Sigma_{1}$ has the closed-loop dynamics of the subnetworks. Therefore, we inspect that

[TABLE]

Additionally, we observe that for $i=2,\dots,N$, we have

[TABLE]

provided that $\phi_{\mathrm{nn}}$ is the performance function computed using matrices in (69).

Appendix I

Proof of Theorem 10: If $\mathbf{K}_{2}=\alpha\mathbf{K}_{1}$, then we can consider the network of network to be a single network with feedback gain $\mathbf{K}_{1}$, over a graph $\mathcal{G}_{3}=\mathcal{G}_{1}\cup\mathcal{G}_{2}$. The weights of links in $\mathcal{G}_{1}$ are preserved, while the weights of the links in $\mathcal{G}_{2}$ are scaled by $\alpha$. Hence, if we increase the weights in the higher level network (equivalently, the eigenvalues of $\mathbf{L}_{2}$), at some point the second smallest eigenvalue of equivalent Laplacian $\mathbf{L}_{3}$ will pass $\tilde{\lambda}(\mathbf{K}_{1})<\infty$.

Appendix J

Proof of Corollary 1: Following the same lines as in the proof of Theorem 10, if the minimum connectivity threshold is zero, for any choice of $\mathbf{K}_{2}=\alpha\mathbf{K}_{1}$, the network with a single equivalent graph and feedback gain $\mathbf{K}_{1}$ has zero minimum connectivity threshold, while the equivalent Laplacian would always have a nonzero $\lambda_{2}$. Hence, the connectivity threshold in terms of the eigenvalues of $\mathbf{L}_{2}$ is zero as well.

Appendix K

Proof of Theorem 11: First we proof an inequality that is an extension of one in [29] in the case of $f(\lambda)=\lambda^{\alpha}$ for $\alpha\notin[0,1]$. For a continuously differentiable convex function $f(x)$ and a Laplacian $\mathbf{L}$ with $M$ edges and maximum degree $\Delta$, we show that

[TABLE]

and the equality holds if and only if $\mathcal{G}$ is complete or star. The steps provided in the proof of this lemma are essentially the same steps reported for Theorem 3 in [29] (only for power functions). Since $f$ is convex and continuous, we use Jensen’s inequality to write

[TABLE]

where if the function $f(\lambda)$ is not affine, then the equality holds if and only if $\lambda_{1}=\dots=\lambda_{N-1}$. This implies we can write

[TABLE]

where the auxiliary function $s(x)$ is defined as

[TABLE]

Because $f(x)$ is continuously differentiable, so is $s(x)$ and

[TABLE]

The function $f(x)$ is convex, thus $f^{\prime}(x)$ is nondecreasing. Then, $s(x)$ is strictly increasing, since for any $x\geq 2M/(N-1)$

[TABLE]

In an unweighted graph, $\lambda_{N}\geq 1+\Delta\geq{2M}/(N-1)$ (see [29] and also [30]). Therefore,

[TABLE]

which proves 115. Applying this on a convex $\phi(\lambda)$, (85) is followed. The equality holds if and only if $\lambda_{2}=\dots=\lambda_{N-1}$ and $\lambda_{N}=1+\Delta$, which happens if and only if $\mathcal{G}$ is either complete or star (again, see both [29] and [30]).

Appendix L

Proof of Theorem 12: Consider any convex function $f$. We start from Jensen’s inequality in the form of

[TABLE]

Because the eigenvalues sum to $2W$, replacing $f$ with $\phi$ gives us the inequality (86). The equality holds if and only if $\lambda_{2}=\dots=\lambda_{N}$ that happens if and only if $\mathcal{G}$ is complete graph with identical weights.

Appendix M

Proof of Theorem 13: For an unweighted path graph

[TABLE]

We define the equidistant partition of interval $[1/N,1]$ as

[TABLE]

We define the following quantities for each interval:

[TABLE]

They induce the following summations

[TABLE]

where $\rho({\mathbf{L},\mathbf{K}})=S_{N}$. These sums imply the natural ordering

[TABLE]

Moreover, compared to $\Gamma_{N}$, we observe that

[TABLE]

We bring a lemma whose proof is given in the next appendix.

Lemma 1**.**

For a rational function $\phi(\lambda)$ that is bounded for any $\lambda\in(0,\infty)$, ${\overline{\phi}_{i,N}}/{\underline{\phi}_{i,N}}\leq\delta_{\phi},$ uniformly over $i$ and $N$ for some $\delta_{\phi}>0$ depending on $\phi$.

We can apply Lemma 1, since $\tilde{\lambda}(\mathbf{K})=0$. As a results, we find that

[TABLE]

This means that $\Gamma_{N}$ and $S_{N}$ are both bounded according to

[TABLE]

If we combine these two inequalities, we find that

[TABLE]

If $\phi(\lambda)$ is bounded, then one deduces that

[TABLE]

Therefore, we can write the following two inequalities

[TABLE]

[TABLE]

Thus, we can write

[TABLE]

For unweighted cycle graphs, the Laplacian eigenvalues are

[TABLE]

Without loss of generality, for deriving the scaling purposes, we may assume that $N$ is $odd$. Then, we will have $(N-1)/2$ distinct values for the eigenvalues of $\mathbf{L}$, where each value is repeated exactly twice. Moreover, we can write

[TABLE]

This time, we need to partition $[1/N,1/2]$ and proceed with identical steps to find out that $2\int_{1/N}^{1/2}\phi\left(2-2\cos\left(2\pi x\right)\right)\leavevmode\nobreak\ dx,$ does the same job in the case of cycle graphs. If we replace $2x\rightarrow x$, the factor $2$ is canceled. This means that (94) and (96) upon replacement of $\Gamma_{N}$ with $\Psi_{N}$ are achieved, where

[TABLE]

Since $\Gamma_{N}=\Theta(\Psi_{N})$, the same conclusion apply for the networks over cycle graphs as well.

Appendix N

Proof of Lemma 1: Let us define $\chi:=\log(2-2\cos(\pi x)),$ and$\varphi(\chi):=\log(\phi(2-2\cos(\pi x))).$ Denote the order of the pole of $\phi(x)$ at $x=0$ by $\alpha\in\mathbb{Z}_{+}$. This means that we can decompose $\varphi$ according to

[TABLE]

for some strictly positive function $\hat{\varphi}(.)$ that is bounded and rational. We can write this in terms of $\chi$ as follows

[TABLE]

In the interval of interest, $\log(.)$, $\hat{\varphi}(.)$ (a bounded and positive rational function) and $\exp(.)$ are all Lipschitz continuous, so is their composition $\log\left(\hat{\varphi}(\exp(\chi))\right)$. This implies that $\varphi(\chi)$ is Lipschitz-continuous. Hence, for $\chi_{1}=\log(2-2\cos(\pi x_{1}))$ and $\chi_{2}=\log(2-2\cos(\pi x_{2}))$, the corresponding Lipschitz continuity inequality will be

[TABLE]

for some Lipschitz constant $\delta_{\varphi}\geq 0$. This is equivalent to

[TABLE]

This means that alternatively we may write

[TABLE]

Let us define

[TABLE]

Since $2-2\cos(\pi x)$ is increasing in $\mathcal{P}_{i}(N)$, for $x_{1},x_{2}\in\mathcal{P}_{i}(N)$

[TABLE]

Moreover, one can find that $h_{N}(x)$ is decreasing for any $x\in[1/N,(N-1)/N]$. Thus, we can write

[TABLE]

We can see note that the right hand side of (120) is

[TABLE]

Computing its formal derivative with respect to $N$, we get

[TABLE]

Thus, its supremum should be evaluated based on the limit

[TABLE]

Let us take the maximum of the left hand side of (119) over $x_{1},x_{2}\in\mathcal{P}_{i}(N)$ and combine it with (121). We conclude that

[TABLE]

which completes the proof.

Appendix O

Proof of Corollary 3: Because $\Gamma_{N}$ and $\Psi_{N}$ scale similarly with respect to $N$, the first part of the claim follows. If the performance function is bounded, then both of them become the same quantity, because we can replace the lower bound of these two integrals with a common limit of [math].

Appendix P

Proof of Result on Input Measures.

We can see that

[TABLE]

We can further write

[TABLE]

Because $\mathbf{M}_{N}\mathbf{L}=\mathbf{L}\mathbf{M}_{N}=\mathbf{L}$, we can write

[TABLE]

If we take the expected value and tend the time to infinity, using the same lines as the proof of Theorem 1, we find that

[TABLE]

Let us take the trace from the both sides. We find that

[TABLE]

Because $\mathbf{U}^{T}\mathbf{L}^{2}\mathbf{U}=\mathbf{\Lambda}^{2}$, we conclude that

[TABLE]

Let us move $\mathbf{KH}$ to the left hand side of the trace arguments. The claim is followed. ∎

Appendix Q:

Additional Details of Examples

Details of Example 1: For networks with nodal dynamics $\mathfrak{s}_{1}$, the noiseless dynamics of $\Sigma_{i}$ are

[TABLE]

that are asymptotically stable if $a-\lambda_{i}k<0$. This confirms $\tilde{\lambda}(\mathbf{K})=\max\left(-{a}/{k},0\right)=0$. The solution to (19) is ${\mathbf{P}}={1}/{2(k\lambda+a)},$ which result in the claimed form for $\phi$. We can see that for $\lambda>\tilde{\lambda}(\mathbf{K})=0$, it is strictly convex and strictly decreasing. The dynamics of the subsystem $\Sigma_{i}$ for nodal dynamics $\mathfrak{s}_{2}$ without disturbance and noise are

[TABLE]

The corresponding characteristic polynomial is

[TABLE]

which is a stable polynomial if and only if $a_{1}+k_{2}\lambda>0,\text{ and }a_{2}+k_{1}\lambda>0,$ that imply $\tilde{\lambda}(\mathbf{K})=\max\left(-{a_{1}}/{k_{2}},-{a_{2}}/{k_{1}},0\right)=0$. Using (19), we get that

[TABLE]

Substitution of this matrix into (18) gives us the results of the table. Now, noting that

[TABLE]

that is sum of two strictly convex and strictly decreasing functions after their negative poles. Those poles are less than or equal to $\tilde{\lambda}(\mathbf{K})$. Hence, the $\phi(\lambda)$ is strictly decreasing and strictly convex in the claimed domain.

For the double integrator with observer, note that

[TABLE]

Observe that its characteristic polynomial is $p_{\lambda}(s)=s^{2}+f_{1}\lambda s+f_{2}\lambda,$ which is stable if and only if $f_{1},f_{2}>0$.

Details of Example 3: For the network of triple-integrators, the characteristic polynomial $\Sigma_{i}$ is

[TABLE]

The stability requires $k_{i}>0$ and

[TABLE]

The solution to for ${\mathbf{P}}(\lambda,\mathbf{K})$ from the Lyapunov equation is shown in Table III. The formula for the performance function then is followed by computing $\mathrm{Tr}(\mathbf{C}^{T}\mathbf{P}\mathbf{C})$.The performance function in this case is strictly decreasing and convex.

Details of Example 5: The realization of the agents is

[TABLE]

The characteristic polynomial of the systems in this case becomes

[TABLE]

(see also [10]). Based on Routh-Hurwitz criteria, for an unbounded stability region, we should impose the following restrictions on $\mathbf{K}$. $k_{1},k_{2}>0,\text{ and }k_{3}\geq 0.$ Moreover, we need the inequality

[TABLE]

from which we may infer the bicriteria definition for $\tilde{\lambda}(\mathbf{K})$ in (60). To find the performance functions, we find ${\mathbf{P}}(\lambda,\mathbf{K})$ which is

[TABLE]

which lets us compute the performance function. If $k_{3}>0$, the function $\phi(\lambda)$ is a positive combination of

[TABLE]

Moreover, $f_{1}$ and $f_{2}$ are products of functions that are strictly convex and decreasing for $\lambda>\tilde{\lambda}(\mathbf{K})$ according to

[TABLE]

Thus, the performance function is this case is also strictly convex and strictly decreasing for $\lambda>\tilde{\lambda}(\mathbf{K})$.

Details of Example 6: The input-output transfer function is $H(s)=(s-\zeta)/{s^{2}}$ with a right-hand plane zero at $\zeta>0$. To evaluate $\mathbf{P}_{0}$, we use a method suggested by [31]. First, we decompose the transfer function according to $H=H_{1}H_{2}$ where the components are

[TABLE]

The transfer function $H_{1}$ has the balanced realization 777A minimal realization of a stable transfer matrix is called balanced if its controllability and observability Gramians are diagonal and equal (such a realization exists for a stable transfer matrix).

[TABLE]

while $H_{2}$ has a stabilizable and detectable realization

[TABLE]

Suppose that the factorized realizations have the state vectors $X$ and $x$, respectively. Then, we can show that they are related based on

[TABLE]

Then, the reference shows that

[TABLE]

Therefore, we get $\mathbf{P}_{0}=\mathrm{diag}\left(2\zeta,0\right).$

Details of Example 8: We can see that $\tilde{\mathbf{E}}=\tilde{\mathbf{C}}=\mathbf{I}_{m}$ and

[TABLE]

The solution to the Lyapunov equation in this case is

[TABLE]

Now, similar to the previous example, we can see that

[TABLE]

Details of Example 9: For subnetworks of single-integrators over $\mathcal{G}_{1}$ that is complete, $\tilde{\mathbf{E}}=\tilde{\mathbf{C}}=\mathbf{I}_{m}$ and

[TABLE]

We can verify that the solution to the Lyapunov equation is

[TABLE]

Then, we can write

[TABLE]

For double-integrators over complete graph modules, similar expressions for the matrices $\tilde{\mathbf{A}},$ $\tilde{\mathbf{B}}$, and $\tilde{\mathbf{C}}$ holds, while the solution to the Lyapunov equation in this case is shown in Table IV. Because the output of the double-integrator is on the first state, we can write

[TABLE]

This proves the claims in the example.

Details of Continuance of Example 1: First, we prove that we can replace $\Gamma_{N}$ in Theorem 13 with

[TABLE]

Considering $\lambda=2-2\cos(\pi x)$, we get that

[TABLE]

Given $\lambda=2-2\cos(\pi x)$, $d\lambda={2\pi}\sqrt{\lambda-\lambda^{2}/4}\leavevmode\nobreak\ dx,$ and

[TABLE]

that are integral limits of interest.

Now, if $a=0$, for the single integrators

[TABLE]

we can compute $\Gamma_{N}$ for $\phi(\lambda)$ as

[TABLE]

Now, if $a>0$, then we need the integral

[TABLE]

This implies that $\Gamma_{N}$ for $\phi(\lambda)$ in this case satisfies

[TABLE]

For $\mathfrak{s}_{2}$ agents with $a_{0}=a_{1}=0$ we need

[TABLE]

Now, we compute $\Gamma_{N}$ for $\phi(\lambda)$ as follows

[TABLE]

Details of Continuance of Example 4: In this case similar computations reveals that for $\phi(\lambda)$

[TABLE]

Details of the Continuance of Example 8: In this case,

[TABLE]

Thus, based on the formula for the performance of the network of networks in (81), the claim is followed.

Details of Example 11: The state space matrices of the aircraft model are borrowed from [24] are given below.

[TABLE]

The result of feedback gain design is

[TABLE]

For the case of observer-based relative output feedback, the following value of $\mathbf{F}$ gives us depicted performance functions.

[TABLE]