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0.6pt,line join=round,line cap=round] ( 37.22, 33.34) rectangle
(232.99,175.17); [clip] ( 0.00, 0.00) rectangle (238.49,180.67);
drawColorgray0.30 [text=drawColor,anchor=base west,inner sep=0pt, outer
sep=0pt, scale= 0.88] at ( 18.34, 36.02) 10; [text=drawColor,anchor=base
west,inner sep=0pt, outer sep=0pt, scale= 0.62] at ( 27.13, 39.62) -6;
[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.88] at
( 18.34, 57.51) 10; [text=drawColor,anchor=base west,inner sep=0pt, outer
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[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.62] at
( 27.13, 82.59) -4; [text=drawColor,anchor=base west,inner sep=0pt, outer
sep=0pt, scale= 0.88] at ( 18.34,100.48) 10; [text=drawColor,anchor=base
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[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.62] at
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west,inner sep=0pt, outer sep=0pt, scale= 0.62] at ( 29.19,168.55) 0; [clip] (
0.00, 0.00) rectangle (238.49,180.67); drawColorgray0.20 [draw=drawColor,line
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33.34); [draw=drawColor,line width= 0.6pt,line join=round] (105.44, 30.59) –
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30.59) – (135.10, 33.34); [draw=drawColor,line width= 0.6pt,line join=round]
(164.77, 30.59) – (164.77, 33.34); [draw=drawColor,line width= 0.6pt,line
join=round] (194.43, 30.59) – (194.43, 33.34); [draw=drawColor,line width=
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west,inner sep=0pt, outer sep=0pt, scale= 0.88] at ( 40.18, 20.84) 10;
[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.62] at
( 48.97, 24.44) 0; [text=drawColor,anchor=base west,inner sep=0pt, outer
sep=0pt, scale= 0.88] at ( 69.84, 20.84) 10; [text=drawColor,anchor=base
west,inner sep=0pt, outer sep=0pt, scale= 0.62] at ( 78.64, 24.44) 1;
[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.88] at
( 99.50, 20.84) 10; [text=drawColor,anchor=base west,inner sep=0pt, outer
sep=0pt, scale= 0.62] at (108.30, 24.44) 2; [text=drawColor,anchor=base
west,inner sep=0pt, outer sep=0pt, scale= 0.88] at (129.16, 20.84) 10;
[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.62] at
(137.96, 24.44) 3; [text=drawColor,anchor=base west,inner sep=0pt, outer
sep=0pt, scale= 0.88] at (158.83, 20.84) 10; [text=drawColor,anchor=base
west,inner sep=0pt, outer sep=0pt, scale= 0.62] at (167.63, 24.44) 4;
[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.88] at
(188.49, 20.84) 10; [text=drawColor,anchor=base west,inner sep=0pt, outer
sep=0pt, scale= 0.62] at (197.29, 24.44) 5; [text=drawColor,anchor=base
west,inner sep=0pt, outer sep=0pt, scale= 0.88] at (218.15, 20.84) 10;
[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.62] at
(226.95, 24.44) 6; [clip] ( 0.00, 0.00) rectangle (238.49,180.67);
drawColorRGB0,0,0 [text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt,
scale= 1.10] at (135.10, 8.01) $k_{in}$; [clip] ( 0.00, 0.00) rectangle
(238.49,180.67); drawColorRGB0,0,0 [text=drawColor,rotate=
90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.10] at (
13.08,104.26) $\Pr[k_{in}]$;


$\displaystyle\begin{bmatrix}Y&\mathbf{I}\\\ \mathbf{I}&X\end{bmatrix}>0,$
(134)
$\displaystyle\begin{bmatrix}Y&\mathbf{I}&YA^{\top}+C_{n}^{\top}B^{\top}&A_{n}^{\top}\\\
\star&X&A^{\top}+C^{\top}D_{n}^{\top}B^{\top}&A^{\top}X+C^{\top}B_{n}^{\top}\\\
\star&\star&Y&\mathbf{I}\\\ \star&\star&\star&X\end{bmatrix}>0,$ (135)

and $A_{c},B_{c},C_{c},D_{c}$ (132) are found by CoVs (35) and (38).

###### Proof.

Applying Prop. 19 to (133) give the LMI conditions necessary and sufficient
for the stabilization of (133) as: $\exists P>0$ such that

$\begin{bmatrix}P&\bar{A}^{\top}P\\\ \star&P\end{bmatrix}>0.$ (136)

Using the CoVs in (34) and (37), the LMI $P>0$ can be transformed to (134).
Finally, (136) can be transformed to (135) by applying Lm. 2 with
$\text{diag}(\Pi_{x}^{\top}P^{-1},\Pi_{x}^{\top}P^{-1})$ and substituting from
(36) as:

$\displaystyle\begin{bmatrix}P&\bar{A}^{\top}P\\\
\star&P\end{bmatrix}>0\iff\begin{bmatrix}\Pi_{x}^{\top}P^{-1}\Pi_{x}&\Pi_{x}^{\top}P^{-1}\bar{A}^{\top}\Pi_{x}\\\
\star&\Pi_{x}^{\top}P^{-1}\Pi_{x}\end{bmatrix}>0$
$\displaystyle\iff\begin{bmatrix}\Pi_{y}^{\top}\Pi_{x}&\Pi_{y}^{\top}\bar{A}^{\top}\Pi_{x}\\\
\star&\Pi_{y}^{\top}\Pi_{x}\end{bmatrix}>0\iff\mbox{\eqref{Eq:Pr:DTStabilizationUnderDOF2}}.$

Note that the last step above results from (40). ∎

####  VI-D2 $(Q,S,R)$-Dissipativation

The following proposition provides an LMI condition that leads to synthesize a
DOF controller (132) such that the closed-loop system (133) is
$(Q,S,R)$-dissipative from $w(t)$ to $z(t)$.

######  Proposition 34.

Under $D=\mathbf{0}$, the closed-loop DT-LTI system (133) is
$(Q,S,R)$-dissipative (with $-Q>0$, $R=R^{\top}$) from $w(t)$ to $z(t)$ iff
$\exists X,Y>0$ and $A_{n},B_{n},C_{n},D_{n}$ such that

$\begin{bmatrix}Y&\mathbf{I}\\\ \mathbf{I}&X\end{bmatrix}>0\mbox{ and
\eqref{Eq:Pr:DTDissipativationUsingDOF2}},$ (137)

and $A_{c},B_{c},C_{c},D_{c}$ (132) are found by CoVs (35) and (38).

###### Proof.

The proof starts with applying Prop. 20 to (133) to obtain the LMI conditions
necessary and sufficient for the $(Q,S,R)$-dissipativation of (133) as:
$\exists P>0$ such that

$\begin{bmatrix}P&\bar{C}^{\top}S&\bar{A}^{\top}P&\bar{C}^{\top}\\\
\star&\bar{D}^{\top}S+S^{\top}\bar{D}+R&\bar{B}^{\top}P&\bar{D}^{\top}\\\
\star&\star&P&\mathbf{0}\\\ \star&\star&\star&-Q^{-1}\end{bmatrix}>0.$ (138)

Similar to the proof of Prop. 33, using the CoVs in (34) and (37), the LMI
$P>0$ can be transformed to (137). To obtain (142) from (138), first, Lm. 2 is
applied with
$\text{diag}(\Pi_{x}^{\top}P^{-1},\mathbf{I},\Pi_{x}^{\top}P^{-1},\mathbf{I}\\}$
and then the results are substituted using (36) to obtain:

$\begin{bmatrix}\Pi_{y}^{\top}\Pi_{x}&\Pi_{y}^{\top}\bar{C}^{\top}S&\Pi_{y}^{\top}\bar{A}^{\top}\Pi_{x}&\Pi_{y}^{\top}\bar{C}^{\top}\\\
\star&\mathcal{H}_{s}(\bar{D}^{\top}S)+R&\bar{B}^{\top}\Pi_{x}&\bar{D}^{\top}\\\
\star&\star&\Pi_{y}^{\top}\Pi_{x}&\mathbf{0}\\\
\star&\star&\star&-Q^{-1}\end{bmatrix}>0.$ (139)

Finally, the above matrix inequality can be transformed to get the LMI in
(142) using (40). ∎

####  VI-D3 $\mathcal{H}_{2}$-Optimal Control

The goal of $\mathcal{H}_{2}$-optimal control here is to synthesize a DOF
controller (132) that minimizes the $\mathcal{H}_{2}$-norm of the closed-loop
system (133) from $w(t)$ to $z(t)$.

######  Proposition 35.

Under $D=\mathbf{0}$, the $\mathcal{H}_{2}$-optimal DOF controller (132) for
the closed-loop system (133) is found by solving the LMI:

$\displaystyle\min_{\begin{subarray}{c}X,Y,Q,\gamma\\\
A_{n},B_{n},C_{n},D_{n}\end{subarray}}$ $\displaystyle\gamma$ (140) sub. to:
$\displaystyle X>0,\ Y>0,\ Q>0,\ \gamma>0,\
\mbox{\eqref{Eq:Pr:DTH2ControlUnderDOF2}},$
$\displaystyle\mbox{tr}\left(Q\right)<\gamma^{2},$

and $A_{c},B_{c},C_{c},D_{c}$ (132) are found by CoVs (35) and (38).

###### Proof.

The proof follows similar steps as that of Prop. 34. ∎

####  VI-D4 $\mathcal{H}_{\infty}$-Optimal Control

The goal of $\mathcal{H}_{\infty}$-optimal control is to synthesize a DOF
controller (132) that minimizes the $\mathcal{H}_{\infty}$-norm of the closed-
loop system (133) (from $w(t)$ to $z(t)$).

######  Proposition 36.

Under $D=0$, the $\mathcal{H}_{\infty}$-optimal DOF controller (132) for the
closed-loop system (133) is found by solving the LMI:

$\displaystyle\min_{\begin{subarray}{c}X,Y,\gamma\\\
A_{n},B_{n},C_{n},D_{n}\end{subarray}}$ $\displaystyle\ \gamma$ (141) sub. to:
$\displaystyle\ X>0,Y>0,\gamma>0,\eqref{Eq:Pr:DTHInfControlUnderDOF2}.$

and $A_{c},B_{c},C_{c},D_{c}$ (132) are found by CoVs (35) and (38).

###### Proof.

The proof follows similar steps as that of Prop. 34. ∎

An outline of all the established theoretical results in this section can be
found in Tab. IV.

$\begin{bmatrix}Y&\mathbf{I}&(YG^{\top}+C_{n}^{\top}H^{\top})S&YA^{\top}+C_{n}^{\top}B^{\top}&A_{n}^{\top}&YG^{\top}+C_{n}^{\top}H^{\top}\\\
\star&X&(G^{\top}+C^{\top}D_{n}^{\top}H^{\top})S&A^{\top}+C^{\top}D_{n}^{\top}B^{\top}&A^{\top}X+C^{\top}B_{n}^{\top}&G^{\top}+C^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\mathcal{H}_{s}((J^{\top}+F^{\top}D_{n}^{\top}H^{\top})S)+R&E^{\top}+F^{\top}D_{n}^{\top}B^{\top}&E^{\top}X+F^{\top}B_{n}^{\top}&J^{\top}+F^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\star&Y&\mathbf{I}&\mathbf{0}\\\
\star&\star&\star&\star&X&\mathbf{0}\\\
\star&\star&\star&\star&\star&-Q^{-1}\end{bmatrix}>0$ (142)

$\displaystyle\begin{bmatrix}Y&\mathbf{I}&YA^{\top}+C_{n}^{\top}B^{\top}&A_{n}^{\top}&YG^{\top}+C_{n}^{\top}H^{\top}\\\
\star&X&A^{\top}+C^{\top}D_{n}^{\top}B^{\top}&A^{\top}X+C^{\top}B_{n}^{\top}&G^{\top}+C^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&Y&\mathbf{I}&\mathbf{0}\\\ \star&\star&\star&X&\mathbf{0}\\\
\star&\star&\star&\star&\mathbf{I}\end{bmatrix}>0\ \ \mathbf{and}$ (143)
$\displaystyle\begin{bmatrix}Q&E^{\top}+F^{\top}D_{n}^{\top}B^{\top}&E^{\top}X+F^{\top}B_{n}^{\top}&J^{\top}+F^{\top}D_{n}^{\top}H^{\top}\\\
\star&Y&\mathbf{I}&\mathbf{0}\\\ \star&\star&X&\mathbf{0}\\\
\star&\star&\star&\mathbf{I}\end{bmatrix}>0$

$\begin{bmatrix}Y&\mathbf{I}&AY+BC_{n}&A+BD_{n}C&E+BD_{n}F&\mathbf{0}\\\
\star&X&A_{n}&XA+B_{n}C&XE+B_{n}F&\mathbf{0}\\\
\star&\star&Y&\mathbf{I}&\mathbf{0}&YG^{\top}+C_{n}^{\top}H^{\top}\\\
\star&\star&\star&X&\mathbf{0}&G^{\top}+C^{\top}D_{n}\top H^{\top}\\\
\star&\star&\star&\star&\gamma\mathbf{I}&J^{\top}+F^{\top}D_{n}^{\top}H^{\top}\\\
\star&\star&\star&\star&\star&\gamma\mathbf{I}\end{bmatrix}>0$ (144)

TABLE IV: Summary of Theoretical Results (DT-LTI Systems).

Proposition # (For DT-LTI Systems) | Concept
---|---
Stability |  |  $(Q,S,R)$-
---
Dissipativity
|  $\mathcal{H}_{2}$-
---
Norm
|  $\mathcal{H}_{\infty}$-
---
Norm
Task |  | DT-LTI System
---
Analysis
19 | 20 | 21 | 22
| FSF Controller
---
Synthesis
25,23 | 26 | 27 | 28
| Observer
---
Design
29,24 | 30 | 31 | 32
| DOF Controller
---
Synthesis
33 | 34 | 35 | 36

##  VII The Discrete-Time Networked System (DTNS)

This section briefly provides the unique details of the considered discrete-
time networked system (DTNS).

###  VII-A Subsystems of the DTNS

####  VII-A1 Dynamics

We consider the dynamics of the $i$th subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$ of the DTNS to be

$\displaystyle x_{i}(t+1)=$
$\displaystyle\sum_{j\in\bar{\mathcal{E}}_{i}}A_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}B_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}E_{ij}w_{j}(t),$
(145a) $\displaystyle y_{i}(t)=$
$\displaystyle\sum_{j\in\bar{\mathcal{E}}_{i}}C_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}D_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}F_{ij}w_{j}(t),$
(145b)

where now $t\in\mathbb{N}$.

####  VII-A2 Local Controllers and Observers

As mentioned before, one main objective of this paper is to design local
controllers/observers at the subsystems of the considered DTNS in a
decentralized manner. In this setting, at a subsystem $\Sigma_{i}$:

  1. 1.

a local FSF controller may take the form:

$u_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}K_{ij}x_{j}(t);$ (146)

  2. 2.

a local Luenberger observer may take the form:

$\hat{x}_{i}(t+1)=\sum_{j\in\bar{\mathcal{E}}_{i}}\hat{A}_{ij}\hat{x}_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}\hat{B}_{ij}u_{j}+\sum_{j\in\bar{\mathcal{E}}_{i}}L_{ij}y_{j}(t);$
(147)

  3. 3.

a local DOF controller may take the form:

$\displaystyle\zeta_{i}(t+1)=\sum_{j\in\bar{\mathcal{E}}_{i}}A_{c,ij}\zeta_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}B_{c,ij}y_{j}(t),$
(148) $\displaystyle
u_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}C_{c,ij}\zeta_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}D_{c,ij}y_{j}(t).$

####  VII-A3 Local Performance Metrics

Recall that a pre-defined local performance metric is required at each
subsystem $\Sigma_{i}$ when designing local controllers and observers in a
$(Q,S,R)$-dissipative or $\mathcal{H}_{2}/\mathcal{H}_{\infty}$-optimal sense.
Therefore, identically to the continuous-time case, at a subsystem
$\Sigma_{i}$, we use the following pre-defined local performance metrics:

  1. 1.

For local FSF controller design:

$y_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}C_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}D_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}F_{ij}w_{j}(t);$
(149)

  2. 2.

For local Luenberger observer design:

$z_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}G_{ij}(x_{j}(t)-\hat{x}_{j}(t))+\sum_{j\in\bar{\mathcal{E}}_{i}}J_{ij}w_{j}(t);$
(150)

  3. 3.

For local DOF controller design:

$z_{i}(t)=\sum_{j\in\bar{\mathcal{E}}_{i}}G_{ij}x_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}H_{ij}u_{j}(t)+\sum_{j\in\bar{\mathcal{E}}_{i}}J_{ij}w_{j}(t).$
(151)

###  VII-B The DTNS

####  VII-B1 Dynamics

By stating (145) for all $i\in\mathbb{N}_{N}$, we can obtain the dynamics of
the networked system $\mathcal{G}_{N}$ as

$\displaystyle x(t+1)=$ $\displaystyle\ Ax(t)+Bu(t)+Ew(t),$ (152a)
$\displaystyle y(t)=$ $\displaystyle\ Cx(t)+Du(t)+Fw(t),$ (152b)

similar to (59) except for the fact that now time $t\in\mathbb{N}$.

####  VII-B2 Controllers and Observers

By composing each local controller/observer forms in (146),(147), (148) for
all $i\in\mathbb{N}_{N}$, we can respectively obtain the network level (i.e.,
global):

  1. 1.

FSF controller

$u(t)=Kx(t);$ (153)

  2. 2.

Luenberger observer

$\hat{x}(t+1)=\hat{A}x(t)+\hat{B}u(t)+Ly(t);$ (154)

  3. 3.

DOF controller

$\displaystyle\zeta(t+1)=A_{c}\zeta(t)+B_{c}y(y),$ (155) $\displaystyle
u(t)=C_{c}\zeta(t)+D_{c}y(y),$

where $K,\hat{A},\hat{B},L,A_{c},B_{c},C_{c},D_{c}$ are all $N\times N$ block
matrices comprised of the corresponding local design parameters.

####  VII-B3 Performance Metrics

Similarly, by composing each pre-defined local controller/observer performance
metric forms in (149),(150),(151) for all $i\in\mathbb{N}_{N}$, we can
respectively obtain the global performance metrics considered for:

  1. 1.

FSF controller design as:

$y(t)=Cx(t)+Dy(t)+Fw(t);$ (156)

  2. 2.

Luenberger observer design as:

$z(t)=G(x(t)-\hat{x}(t))+Jw(t);$ (157)

  3. 3.

DOF controller design as:

$z(t)=Gx(t)+Hu(t)+Jw(t).$ (158)

Here also $C,D,F,G,H,J$ are all $N\times N$ block matrices comprised of the
corresponding pre-defined local performance metric parameters (e.g.,
$C=[C_{ij}]_{i,j\in\mathbb{N}_{N}}$).

###  VII-C The Research Problem

Similar to the continuous-time case, the forms of the DTNS (152), global
controllers/observers (153)-(155) and global performance metrics (156)-(158)
are respectively identical to the general DT-LTI system (e.g., (117)),
controllers/observers (e.g., (123),(132)) and performance metrics (e.g.,
(131),(127)) considered in Sec. VI. Therefore, all the LMI-based control
solutions (Prop. 19-36) discussed in Sec. VI are directly applicable for the
DTNS (152). Moreover, as pointed out in Rm. 2, we have formulated these LMI-
based control solutions such that they can be applied in a decentralized
setting (via Alg. 1). In other words, we can use the said LMI-based control
solutions for decentralized analysis and controller/observer synthesis of the
considered DTNS (152).

##  VIII Distributed Analysis and Control Synthesis of DTNS

In this section, parallel to Sec. V, we provide decentralized, compositional
and possibly distributed techniques for different analysis and control
synthesis tasks discussed in Sec. VI for DTNSs introduced in Sec. VII
exploiting the algorithms proposed in Sec. IV. Following the same steps as
before, we have summarized the details of each decentralized technique that we
propose in Tab. III. To provide examples, in the following subsections,
stability and dissipativity related decentralized techniques have been
formally stated as theorems (analogous to Theorems 1-8 provided in Sec. V).
However, note that we omit providing proofs in this section as they can be
obtained by following similar steps to their counterparts in Sec. V.

###  VIII-A Stability Related Decentralized Results

In this subsection, we provide stability based results on:

  1. 1.

decentralized stability analysis,

  2. 2.

decentralized FSF stabilization,

  3. 3.

decentralized stable observer design,

  4. 4.

decentralized DOF stabilization,

that respectively corresponds to Props. 19, 25, 29 and 33 (Tab. III-Rows 1, 7,
11 and 15).

######  Theorem 9.

(Stability analysis) The DTNS (152) under $u(t)=\mathbf{0}$ and
$w(t)=\mathbf{0}$ is stable if at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{9}:\ \ \text{Find}\ \ P_{ii}\ \ \text{such that}\ \ P_{ii}>0,\ \
\tilde{W}_{ii}>0,$ (159)

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
analyzing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with

$W_{ij}=\begin{bmatrix}P_{ii}e_{ij}&A_{ji}^{\top}P_{jj}\\\
P_{ii}A_{ij}&P_{ii}e_{ij}\end{bmatrix}.$ (160)

######  Theorem 10.

(FSF Stabilization) The DTNS (152) (where $B$ is block diagonal) under
$D=\mathbf{0}$, $w(t)=\mathbf{0}$ and local FSF control (146) is stable if at
each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{10}:\ \ \text{Find}\ \ M_{ii},\ \ L_{i}\ \ \text{such that}\ \
M_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (161)

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
enforcing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with

$W_{ij}=\begin{bmatrix}M_{ii}e_{ij}&M_{ii}A_{ji}^{\top}+L_{ji}^{\top}B_{ii}^{\top}\\\
\star&M_{ii}e_{ij}\end{bmatrix}$ (162)

The local FSF controller gains $K_{i}$ are computed using $M_{ii}$ and $L_{i}$
identically to Th. 2 via (86).

######  Theorem 11.

(Stable observer design) For the DTNS (152) (where $C,D$ are block diagonal)
under $w(t)=\mathbf{0}$, the local Luenberger observers (147) render the state
estimation error dynamics (123) stable if at each subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{11}:\ \ \text{Find}\ \ P_{ii},\ \ K_{i}\ \ \text{such that}\ \
P_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (163)

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
enforcing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with

$W_{ij}=\begin{bmatrix}P_{ii}e_{ij}&A_{ji}^{\top}P_{jj}-C_{ii}^{\top}K_{ji}^{\top}\\\
\star&P_{ii}e_{ij}\end{bmatrix}$ (164)

The local Luenberger observer parameters $L_{i},\hat{A}_{i}$ and $\hat{B}_{i}$
(147) are computed using $P_{ii}$ and $K_{i}$ identically to Th. 3 via (89).

######  Theorem 12.

(DOF Stabilization) The DTNS (152) (where $B,C$ are block diagonal) under
$D=\mathbf{0}$, $w(t)=\mathbf{0}$ and local DOF control (148) is stable if at
each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\displaystyle\mathbb{P}_{12}:\ \ \ \ \ \ \text{Find}$ $\displaystyle\ \
X_{ii},\ Y_{ii},\ A_{n,i},\ B_{n,i},\ C_{n,i},\ D_{n,i}$ (165) such that
$\displaystyle\ \ X_{ii}>0,\ Y_{ii}>0,\ \tilde{W}_{ii}^{(1)}>0,\
\tilde{W}_{ii}^{(2)}>0,$

is feasible, where $\tilde{W}_{ii}^{(k)}$ is computed from Alg. 1 (Steps:
3-16) when enforcing $W^{(k)}=[W_{ij}^{(k)}]_{i,j\in\mathbb{N}_{N}}>0$, for
$k=1,2,$ with

$\displaystyle W_{ij}^{(1)}=\begin{bmatrix}Y_{ii}e_{ij}&e_{ij}\\\
e_{ij}&X_{ij}e_{ij}\end{bmatrix},\ \ \ \ W_{ij}^{(2)}=$ (166)
$\displaystyle\begin{bmatrix}Y_{ii}e_{ij}&e_{ij}&Y_{ii}A_{ji}^{\top}+C_{n,ji}^{\top}B_{jj}^{\top}&A_{n,ji}^{\top}\\\
\star&X_{ii}e_{ij}&A_{ji}^{\top}+C_{ii}^{\top}D_{n,ji}^{\top}B_{jj}^{\top}&A_{ji}^{\top}X_{jj}+C_{ii}^{\top}B_{n,ji}^{\top}\\\
\star&\star&Y_{ii}e_{ij}&e_{ij}\\\
\star&\star&\star&X_{ii}e_{ij}\end{bmatrix}.$

The local DOF controller parameters $A_{c,i},B_{c,i},C_{c,i}$ and $D_{c,i}$
are computed using $X_{ii},Y_{ii},A_{n,i},B_{n,i},C_{n,i}$ and $D_{n,i}$
identically to Th. 4 via (92) and (93).

###  VIII-B Dissipativity Related Decentralized Results

In this subsection, analogous to Theorems 9-12, we provide
$(Q,S,R)$-dissipativity based results for:

  1. 1.

decentralized $(Q,S,R)$-dissipativity analysis,

  2. 2.

decentralized FSF $(Q,S,R)$-dissipativation,

  3. 3.

decentralized $(Q,S,R)$-dissipative observer design,

  4. 4.

decentralized DOF $(Q,S,R)$-dissipativation,

that respectively corresponds to Props. 20, 26, 30 and 34 (Tab. III-Rows 2, 8,
12 and 16). In what follows, regarding the given specification matrices
$Q,S,R$, it is assumed that: (1) $Q$ is a block diagonal network matrix, (2)
$-Q>0$, and (3) $R=R^{\top}$.

######  Theorem 13.

(Dissipativity analysis) The DTNS (152) (where $C,D$ are block diagonal) under
$w(t)=\mathbf{0}$ is $(Q,S,R)$-dissipative from $u(t)$ to $y(t)$ if at each
subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{13}:\ \ \text{Find}\ \ P_{ii}\ \ \text{such that}\ \ P_{ii}>0,\ \
\tilde{W}_{ii}>0,$ (167)

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
analyzing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with

$W_{ij}=\begin{bmatrix}P_{ii}e_{ij}&C_{ii}^{\top}S_{ij}&A_{ji}^{\top}P_{jj}&C_{ii}^{\top}e_{ij}\\\
\star&\mathcal{H}_{e}(D_{ii}^{\top}S_{ij})+R_{ij}&B_{ji}^{\top}P_{jj}&D_{ii}^{\top}e_{ij}\\\
\star&\star&P_{ii}e_{ij}&\mathbf{0}\\\
\star&\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}.$ (168)

######  Theorem 14.

(FSF Dissipativation) The DTNS (152) (where $B,C,F$ are block diagonal) under
$D=\mathbf{0}$ and local FSF control (146) is $(Q,S,R)$-dissipative from
$w(t)$ to $y(t)$ if at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the
problem

$\mathbb{P}_{14}:\ \ \text{Find}\ \ M_{ii},\ \ L_{i}\ \ \text{such that}\ \
M_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (169)

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
enforcing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with $W_{ij}$ given in (172).
The local FSF controller gains $K_{i}$ are computed using $M_{ii}$ and $L_{i}$
identically to Th. 2 via (86).

######  Theorem 15.

(Dissipative observer design) For the DTNS (152) (where $C,D,F$ are block
diagonal), the local Luenberger observers (147) with the local performance
metrics (150) (such that $G,J$ in (157) are block diagonal) render the global
state estimation error dynamics (127) $(Q,S,R)$-dissipative from $w(t)$ to
$z(t)$ if at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\mathbb{P}_{15}:\ \ \text{Find}\ \ P_{ii},\ \ K_{i}\ \ \text{such that}\ \
P_{ii}>0,\ \ \tilde{W}_{ii}>0,$ (170)

is feasible, where $\tilde{W}_{ii}$ is computed from Alg. 1 (Steps: 3-16) when
enforcing $W=[W_{ij}]_{i,j\in\mathbb{N}_{N}}>0$ with $W_{ij}$ given in (173).
The local Luenberger observer parameters $L_{i},\hat{A}_{i}$ and $\hat{B}_{i}$
(54) are computed using $P_{ii}$ and $K_{i}$ identically to Th. 3 via (89).

######  Theorem 16.

(DOF Dissipativation) The DTNS (152) (where $B,C,F$ are block diagonal) under
$D=\mathbf{0}$, local DOF control (148) and local performance metrics (151)
(such that $H$ in (158) is block diagonal), i.e., (133), is
$(Q,S,R)$-dissipative (where $S$ is block diagonal) from $w(t)$ to $z(t)$ if
at each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$, the problem

$\displaystyle\mathbb{P}_{16}:\ \ \ \ \ \ \text{Find}$ $\displaystyle\ \
X_{ii},\ Y_{ii},\ A_{n,i},\ B_{n,i},\ C_{n,i},\ D_{n,i}$ (171) such that
$\displaystyle\ \ X_{ii}>0,\ Y_{ii}>0,\ \tilde{W}_{ii}^{(1)}>0,\
\tilde{W}_{ii}^{(2)}>0,$

is feasible, where $\tilde{W}_{ii}^{(k)}$ is computed from Alg. 1 (Steps:
3-16) when enforcing $W^{(k)}=[W_{ij}^{(k)}]_{i,j\in\mathbb{N}_{N}}>0$ for
$k=1,2,$ with $\scriptsize W_{ij}^{(1)}=\begin{bmatrix}Y_{ii}e_{ij}&e_{ij}\\\
e_{ij}&X_{ij}e_{ij}\end{bmatrix}$ and $W_{ij}^{(2)}$ given in (174). The local
DOF controller parameters $A_{c,i},B_{c,i},C_{c,i}$ and $D_{c,i}$ are computed
using $X_{ii},Y_{ii},A_{n,i},B_{n,i},C_{n,i}$ and $D_{n,i}$, identically to
Th. 4 via (92) and (93).

We emphasize that local problems $\mathbb{P}_{9}$-$\mathbb{P}_{16}$ stated
respectively in the theorems 9-16 are LMI problems due to the applicability of
Lm. 1 to simplify the matrix inequality $\tilde{W}_{ii}>0$ in Alg. 1 (Step
14). Consequently, such problems can be solved conveniently and efficiently
using readily available LMI software toolboxes [35]. Moreover, based on the
remaining propositions provided in Sec. VI, a similar set of theorems can be
proposed providing respective decentralized techniques (as summarized in the
respective remaining rows in Tab. III).

$W_{ij}=\begin{bmatrix}M_{ii}e_{ij}&M_{ii}C_{ii}^{\top}S_{ij}&M_{ii}A_{ji}^{\top}+L_{ji}^{\top}B_{jj}^{\top}&M_{ii}C_{ii}^{\top}e_{ij}\\\
\star&\mathcal{H}_{e}(F_{ii}^{\top}S_{ij})+R_{ij}&E_{ji}^{\top}&F_{ii}^{\top}e_{ij}\\\
\star&\star&M_{ii}e_{ij}&\mathbf{0}\\\
\star&\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}$ (172)

$W_{ij}=\begin{bmatrix}P_{ii}e_{ij}&G_{ii}^{\top}S_{ij}&A_{ji}^{\top}P_{jj}-C_{ii}^{\top}K_{ji}^{\top}&G_{ii}^{\top}e_{ij}\\\
\star&\mathcal{H}_{e}(J_{ii}^{\top}S_{ij})+R_{ij}&E_{ji}^{\top}P_{jj}-F_{ii}^{\top}K_{ji}^{\top}&J_{ii}^{\top}e_{ij}\\\
\star&\star&P_{ii}e_{ij}&\mathbf{0}\\\
\star&\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}$ (173)

$\scriptsize
W_{ij}^{(2)}=\begin{bmatrix}Y_{ii}e_{ij}&e_{ij}&(Y_{ii}G_{ji}^{\top}+C_{n,ji}^{\top}H_{jj}^{\top})S_{jj}&Y_{ii}A_{ji}^{\top}+C_{n,ji}^{\top}B_{jj}^{\top}&A_{n,ji}^{\top}&Y_{ii}G_{ji}^{\top}+C_{n,ji}^{\top}H_{jj}^{\top}\\\
\star&X_{ii}e_{ij}&(G_{ji}^{\top}+C_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top})S_{jj}&A_{ji}^{\top}+C_{ii}^{\top}D_{n,ji}^{\top}B_{jj}^{\top}&A_{ji}^{\top}X_{jj}+C_{ii}^{\top}B_{n,ji}^{\top}&G_{ji}^{\top}+C_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top}\\\
\star&\star&\mathcal{H}_{s}((J_{ji}^{\top}+F_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top})S_{jj})+R_{ij}&E_{ji}^{\top}+F_{ii}^{\top}D_{n,ji}^{\top}B_{jj}^{\top}&E_{ji}^{\top}X_{jj}+F_{ii}^{\top}B_{n,ji}^{\top}&J_{ji}^{\top}+F_{ii}^{\top}D_{n,ji}^{\top}H_{jj}^{\top}\\\
\star&\star&\star&Y_{ii}e_{ij}&e_{ij}&\mathbf{0}\\\
\star&\star&\star&\star&X_{ii}e_{ij}&\mathbf{0}\\\
\star&\star&\star&\star&\star&-Q_{ii}^{-1}e_{ij}\end{bmatrix}$ (174)

##  IX Simulation Results

$\begin{aligned} \Sigma_{1}=&\
\begin{cases}\dot{x}_{1}=\begin{bmatrix}0.198&3.412\\\
-3.412&0.198\end{bmatrix}x_{1}+\begin{bmatrix}-0.114&-0.038\\\
-0.038&-0.073\end{bmatrix}x_{2}+\begin{bmatrix}-0.060&-1.032\\\
1.032&-0.060\end{bmatrix}x_{4}+\begin{bmatrix}0.000\\\
0.905\end{bmatrix}u_{1}+\begin{bmatrix}0.000\\\
-0.013\end{bmatrix}w_{1}+\begin{bmatrix}0.000\\\
0.003\end{bmatrix}w_{2}+\begin{bmatrix}-0.001\\\ 0.006\end{bmatrix}w_{4},\\\
y_{1}=\begin{bmatrix}1.114&-2.429\end{bmatrix}x_{1}+\begin{bmatrix}0.000\end{bmatrix}w_{1},\hfill
z_{1}=\begin{bmatrix}1.000&1.000\end{bmatrix}x_{1}+\begin{bmatrix}1.000\end{bmatrix}u_{1}+\begin{bmatrix}1.000\end{bmatrix}w_{1}.\\\
\end{cases}\\\ \Sigma_{2}=&\
\begin{cases}\dot{x}_{2}=\begin{bmatrix}-0.000&-0.001\\\
-0.001&-0.194\end{bmatrix}x_{1}+\begin{bmatrix}1.547&3.164\\\
-3.164&1.547\end{bmatrix}x_{2}+\begin{bmatrix}-0.258&-0.008\\\
-0.008&-0.204\end{bmatrix}x_{4}+\begin{bmatrix}-0.902\\\
0.000\end{bmatrix}u_{2}+\begin{bmatrix}-0.004\\\
-0.010\end{bmatrix}w_{1}+\begin{bmatrix}-0.000\\\
0.021\end{bmatrix}w_{2}+\begin{bmatrix}0.003\\\ 0.002\end{bmatrix}w_{4},\\\
y_{2}=\begin{bmatrix}0.000&1.062\end{bmatrix}x_{2}+\begin{bmatrix}0.002\end{bmatrix}w_{2},\hfill
z_{2}=\begin{bmatrix}1.000&1.000\end{bmatrix}x_{2}+\begin{bmatrix}1.000\end{bmatrix}u_{2}+\begin{bmatrix}1.000\end{bmatrix}w_{2}.\\\
\end{cases}\\\ \Sigma_{3}=&\
\begin{cases}\dot{x}_{3}=\begin{bmatrix}-0.232&-0.070\\\
-0.070&-0.158\end{bmatrix}x_{1}+\begin{bmatrix}-0.096&-0.062\\\
-0.062&-0.085\end{bmatrix}x_{2}+\begin{bmatrix}10.791&5.354\\\
5.354&3.134\end{bmatrix}x_{3}+\begin{bmatrix}-0.074&-0.384\\\
0.384&-0.074\end{bmatrix}x_{4}+\begin{bmatrix}-0.324\\\
-1.406\end{bmatrix}u_{3}+\begin{bmatrix}0.000\\\ -0.007\end{bmatrix}w_{1}\\\
+\begin{bmatrix}-0.002\\\ -0.004\end{bmatrix}w_{2}+\begin{bmatrix}0.000\\\
-0.008\end{bmatrix}w_{3}+\begin{bmatrix}-0.003\\\ -0.001\end{bmatrix}w_{4},\ \
\ \
y_{3}=\begin{bmatrix}1.052&0.759\end{bmatrix}x_{3}+\begin{bmatrix}-0.011\end{bmatrix}w_{3},\
\ \ \
z_{3}=\begin{bmatrix}1.000&1.000\end{bmatrix}x_{3}\begin{bmatrix}1.000\end{bmatrix}u_{3}\begin{bmatrix}1.000\end{bmatrix}w_{3}.\\\
\end{cases}\\\ \Sigma_{4}=&\
\begin{cases}\dot{x}_{4}=\begin{bmatrix}-0.180&-0.066\\\
-0.066&-0.078\end{bmatrix}x_{1}+\begin{bmatrix}1.669&2.302\\\
2.302&3.175\end{bmatrix}x_{4}+\begin{bmatrix}0.000\\\
0.998\end{bmatrix}u_{4}+\begin{bmatrix}-0.006\\\
0.003\end{bmatrix}w_{1}+\begin{bmatrix}-0.018\\\ 0.008\end{bmatrix}w_{4},\\\
y_{4}=\begin{bmatrix}0.629&0.000\end{bmatrix}x_{4}+\begin{bmatrix}0.000\end{bmatrix}w_{4},\hfill
z_{4}=\begin{bmatrix}1.000&1.000\end{bmatrix}x_{4}\begin{bmatrix}1.000\end{bmatrix}u_{4}\begin{bmatrix}1.000\end{bmatrix}w_{4}.\\\
\end{cases}\\\ \Sigma_{5}=&\
\begin{cases}\dot{x}_{5}=\begin{bmatrix}-0.059&0.033\\\
0.033&-0.057\end{bmatrix}x_{4}+\begin{bmatrix}0.058&0.250\\\
0.250&1.074\end{bmatrix}x_{5}+\begin{bmatrix}0.870\\\
-1.461\end{bmatrix}u_{5}+\begin{bmatrix}-0.001\\\
-0.004\end{bmatrix}w_{4}+\begin{bmatrix}-0.006\\\ -0.000\end{bmatrix}w_{5},\\\
y_{5}=\begin{bmatrix}-0.552&-0.750\end{bmatrix}x_{5}+\begin{bmatrix}0.000\end{bmatrix}w_{5},\hfill
z_{5}=\begin{bmatrix}1.000&1.000\end{bmatrix}x_{5}\begin{bmatrix}1.000\end{bmatrix}u_{5}\begin{bmatrix}1.000\end{bmatrix}w_{5}.\\\
\end{cases}\end{aligned}$

(175)

$\displaystyle K_{1}=$ $\displaystyle\
\left\\{K_{11}=\begin{bmatrix}-0.296&-1.523\end{bmatrix}\right\\},\ \ \ \
K_{2}=\left\\{K_{21}=\begin{bmatrix}-0.168&-0.043\end{bmatrix},K_{12}=\begin{bmatrix}-0.173&0.574\end{bmatrix},K_{22}=\begin{bmatrix}5.369&-3.916\end{bmatrix}\right\\},$
(176) $\displaystyle K_{3}=$ $\displaystyle\
\left\\{K_{31}=\begin{bmatrix}-0.083&-0.118\end{bmatrix},K_{32}=\begin{bmatrix}-0.057&-0.067\end{bmatrix},K_{33}=\begin{bmatrix}32.449&5.429\end{bmatrix}\right\\},$
$\displaystyle K_{4}=$ $\displaystyle\
\left\\{K_{41}=\begin{bmatrix}1.023&0.279\end{bmatrix},K_{14}=\begin{bmatrix}-0.973&0.114\end{bmatrix},K_{24}=\begin{bmatrix}-0.234&0.041\end{bmatrix},K_{34}=\begin{bmatrix}0.248&-0.109\end{bmatrix},K_{44}=\begin{bmatrix}-12.974&-8.081\end{bmatrix}\right\\},$
$\displaystyle K_{5}=$ $\displaystyle\
\left\\{K_{54}=\begin{bmatrix}0.035&-0.039\end{bmatrix},K_{55}=\begin{bmatrix}1.753&2.803\end{bmatrix}\right\\}.$

$\displaystyle L_{1}=$ $\displaystyle\ \left\\{L_{11}=\begin{bmatrix}0.306\\\
-0.427\end{bmatrix}\right\\},\ \ \ \
L_{2}=\left\\{L_{21}=\begin{bmatrix}0.060\\\
-0.008\end{bmatrix},L_{12}=\begin{bmatrix}-0.042\\\
-0.207\end{bmatrix},L_{22}=\begin{bmatrix}-3.403&4.596\end{bmatrix}\right\\},$
(177) $\displaystyle L_{3}=$ $\displaystyle\
\left\\{L_{31}=\begin{bmatrix}-0.012\\\
0.043\end{bmatrix},L_{32}=\begin{bmatrix}-0.059\\\
-0.080\end{bmatrix},L_{33}=\begin{bmatrix}12.219&3.616\end{bmatrix}\right\\},$
$\displaystyle L_{4}=$ $\displaystyle\ \left\\{L_{41}=\begin{bmatrix}-0.577\\\
-1.591\end{bmatrix},L_{14}=\begin{bmatrix}-0.640\\\
1.395\end{bmatrix},L_{24}=\begin{bmatrix}-0.409\\\
-0.012\end{bmatrix},L_{34}=\begin{bmatrix}-0.117\\\
0.610\end{bmatrix},L_{44}=\begin{bmatrix}35.602&97.048\end{bmatrix}\right\\},$
$\displaystyle L_{5}=$ $\displaystyle\ \left\\{L_{54}=\begin{bmatrix}-0.094\\\
0.053\end{bmatrix},L_{55}=\begin{bmatrix}0.589&-3.585\end{bmatrix}\right\\}.$

(a) Considered networked system (175) and simulation environment.

(b) Evaluated three distributed controller configurations for the
stabilization of the networked system: (1) FSFC: via local full-state feedback
control, (2) SOFC: via local state observer based feedback control, and (3)
DOFC: via local dynamic output feedback control.

Figure 4: Considered simulation example setup: (a) the networked system and
(b) the controller configurations.

(a)

(b)

(c)

(d)

Figure 5: The used signals as subsystem: (a) inputs
$[u_{i}(t)]_{i\in\mathbb{N}_{N}}$ and (b) disturbances
$[w_{i}(t)]_{i\in\mathbb{N}_{N}}$, and the resulting open-loop signals for
subsystem: (c) outputs $[y_{i}(t)]_{i\in\mathbb{N}_{N}}$ and (d) performances
$[z_{i}(t)]_{i\in\mathbb{N}_{N}}$.

(a) FSFC (centrally derived)
MAO = 1.121

(b) FSFC (decentrally derived)
MAO = 1.063 (Impr.:+5.189%)

(c) SOFC (centrally derived)
MAO = 3.413

(d) SOFC (decentrally derived)
MAO = 2.498 (Impr.:+26.81%)

(e) DOFC (centrally derived)
MAO = 4.227

(f) DOFC (decentrally derived)
MAO = 3.107 (Impr.:+26.49%)

Figure 6: The effect of decentralization: Comparison of output trajectories
$[y_{i}(t)]_{i\in\mathbb{N}_{N}}$ obtained under: centrally (left) vs.
decentrally (right) synthesized stabilizing controllers - for the distributed
controller configurations: FSFC (a,b), SOFC (c,d) and DOFC (e,f) (see Fig.
4(b)). The observed mean absolute output (MAO) values are given in subcaptions
along with the corresponding percentage improvement values.

(a) SOFC (derived:
centrally to stabilize)
MAP = 1.327

(b) SOFC (derived:
centrally to dissipativate)
MAP = 1.246 (Impr.:+6.112%)

(c) SOFC (derived:
decentrally to stabilize)
MAP = 1.155

(d) SOFC (derived:
decentrally to dissipativate)
MAP = 1.015 (Impr.:+12.10%)

(e) DOFC (derived:
centrally to stabilize)
MAP = 16.01

(f) DOFC (derived:
centrally to dissipativate)
MAP = 5.966 (Impr.:+62.74%)

(g) DOFC (derived:
decentrally to stabilize)
MAP = 13.37

(h) DOFC (derived:
decentrally to dissipativate)
MAP = 7.323 (Impr.:+45.21%)

Figure 7: The effect of dissipativation on performance: Comparison of
performance metric trajectories $[z_{i}(t)]_{i\in\mathbb{N}_{N}}$ obtained
under: stabilizing (left) vs. dissipativating (right) controllers -
synthesized both centrally (a,b,e,f) and decentrally (c,d,g,h) for SOFC
(a,b,c,d) and DOFC (e,f,g,h). The observed mean absolute performance (MAP)
values are given in subcaptions along with the corresponding percentage
improvement values.

(a) Output (under centrally
derived stabilizing DOFC)
MAO = 2.978

(b) Output (under centrally
derived dissipativating DOFC)
MAO = 1.890 (Impr.:+36.53%)

(c) Output (under decentrally
derived stabilizing DOFC)
MAO = 2.219

(d) Output (under decentrally
derived dissipativating DOFC)
MAO = 2.370 (Impr.:-6.791%)

(e) Performance (under centrally
derived stabilizing DOFC)
MAP = 28.98

(f) Performance (under centrally
derived dissipativating DOFC)
MAP = 22.70 (Impr.:+21.66%)

(g) Perform. (under decentrally
derived stabilizing DOFC)
MAP = 27.26

(h) Perform. (under decentrally
derived dissipativating DOFC)
MAP = 24.05 (Impr.:+11.81%)

Figure 8: The effect of dissipativation on disturbance rejection: Comparison
of output $[y_{i}(t)]_{i\in\mathbb{N}_{N}}$ and performance metric
$[z_{i}(t)]_{i\in\mathbb{N}_{N}}$ trajectories obtained under: stabilizing
(left) vs. dissipativating (right) controllers - synthesized both centrally
(a,b,e,f) and decentrally (c,d,g,h) for DOFC.

In this section, we provide a collection of numerical results obtained from a
randomly generated networked system, so as to illustrate the applicability of
the proposed: (1) decentralized analysis and control synthesis techniques (in
particular, Theorems. 1, 2, 3 and 4), and (2) $(Q,S,R)$-dissipativity based
centralized (or decentralized) distributed Luenberger observer design and DOF
controller design techniques, i.e., Props. 12 and 16 (or Theorems 7 and 8),
respectively.

Consider the networked dynamical system $\mathcal{G}_{5}$ comprised of five
subsystems $\\{\Sigma_{i}:i\in\mathbb{N}_{5}\\}$ described in (175) that are
interconnected according to the directed network topology shown in Fig. 4(a)
(top-right). Note that, this network topology and each of the subsystem
dynamic models have been generated randomly (using random geometric graphs
[50] and MATLAB “rss($\cdot$)” command, respectively). In particular, using
MATLAB and YALMIP [51], by implementing the said random networked system
generator together with many of the previously proposed centralized and
decentralized analysis and control synthesis techniques (Props. 1, 2, 7, 8,
11, 12, 15, 16 (centralized) and Theorems 1, 5, 2, 6, 3, 7, 4, 8
(decentralized)), we have developed a general software framework for analysis
and control synthesis of arbitrary networked dynamical systems of the form
(59) (available at
https://github.com/shiran27/NetworkedSystemAnalysisAndControl).

When executing different proposed decentralized processes for the considered
networked system $\mathcal{G}_{5}$, here we limit ourselves to the default
subsystem indexing scheme $1-2-3-4-5$. The decentralized stability analysis
process proposed in Th. 1 returns infeasible for $\mathcal{G}_{5}$ \- which
implies the possibility of $\mathcal{G}_{5}$ being unstable. This calls for
stabilizing control synthesis, and as shown in Fig. 4(b), three distributed
controller configurations are considered: (1) using local full-state feedback
control (53) (labeled FSFC), (2) using local state observer (54) based local
state feedback control (labeled SOFC), and (3) using local dynamic output
feedback control (55) (labeled DOFC). Upon synthesizing such local controllers
(either centrally or decentrally, aiming to stabilize or
$(Q,S,R)$-dissipativate), we use MATLAB Simulink (see Fig. 4) to
simulate/assess the closed-loop networked system’s behavior under certain
input and noise (disturbance) processes shown in Figs. 5(a) and 5(b),
respectively. Note also that Figs. 5(c) and 5(d) show the unstable open-loop
output (59) and performance (58) trajectories of the networked system (i.e.,
of the subsystems).

##### Decentralization

Corresponding to the aforementioned three distributed control system
configurations: FSFC, SOFC and DOFC, Fig. 6 shows the observed subsystem
output profiles $[y_{i}(t)]_{i\in\mathbb{N}_{N}}$ under stabilizing
controllers/observers derived: (1) centrally (via Props. 7, 11 and 15, see
Figs. 6(a,c,e)) and (2) decentrally (via Theorems 2, 3 and 4, Figs. 6(b,d,f)).
Based on these observations, it is clear that decentrally designed controllers
performs similar to their centrally designed counter parts. In fact,
interestingly, based on the observed mean absolute output values (MAO,
reported in subcaptions in Fig. 6), it can even be concluded that decentrally
derived controllers result in smoother (less fluctuations) and faster output
trajectories. A probable reason behind this observation may be the emphasis
that decentralized control synthesis has on individual (characteristic) agent
dynamics components (compared to that in centralized control synthesis).
Before moving on, note that, in FSFC and SOFC, based on Theorems 2 and 3,
decentrally derived stabilizing controller and observer gains are given in
(176) and (177), respectively.

##### Dissipativation

In the sequel, by dissipativation, we simply refer to the
$(Q,S,R)$-dissipativation with
$Q=-0.2\mathbf{I},S=\frac{1}{2}\mathbf{I},R=-0.2\mathbf{I}$ (i.e., based on
Rm. 1, strict passivation with input feedforward and output feedback passivity
indices as $\nu=0.2$ and $\rho=0.2$, respectively). As pointed out earlier,
unlike stabilizing control synthesis, dissipativating control synthesis takes
into account the disturbances $w_{i}(t),i\in\mathbb{N}_{N}$ as well as
underlying/interested performance metrics $z_{i}(t)$ (e.g., see (57) and (58))
while also ensuring stability. Therefore, it is reasonable to expect
(hypothesize) dissipativating controllers/observers to provide: (1) better
(lower) performance metric trajectories and (2) better robustness to
disturbances (smoother output and performance profiles), compared to
stabilizing controllers/observers.

In Fig. 7, limiting to distributed control system configurations: SOFC and
DOFC, we test the aforementioned first hypothesis by comparing performance
metric trajectories observed under: (1) a stabilizing controller/observer
(derived centrally using Props. 11 and 15 or decentrally using Theorems 3 and
4), and (2) a dissipativating controller/observer (derived centrally using
Props. 12 and 16 or decentrally using Theorems 7 and 8). The fact that
dissipativating controllers/observers provide superior performance metric
trajectories than stabilizing controllers/observers is evident from the
reported observations in Fig. 7 \- particularly from the provided mean
absolute performance (MAP) values in the subcaptions. We point out that
deriving dissipative distributed full-state feedback controllers (i.e., FSFC),
both centrally and decentrally, turned out to be infeasible for the considered
networked system. We also highlight that performance improvements due to
dissipativation is more prominent when used for distributed dynamic output
feedback control (i.e., DOFC) rather than distributed state observer based
feedback control (i.e., SOFC).

In Fig. 8, limiting to the distributed control system configuration: DOFC, we
test the previously mentioned second hypothesis: dissipativating controllers
have superior disturbance rejection qualities as opposed to stabilizing
controllers. For this purpose, we first superimpose each subsystem disturbance
signal $w_{i}(t),i\in\mathbb{N}_{N}$ shown in Fig. 5(b) with a single square
pulse (of width $3\,s$) occurring between $t\in[20,27]$ as indicated in Fig.
4(a) (the exact form of this pulse is indicative from Figs. 8(e,f,g,h)). The
resulting output and performance trajectories obtained under centrally and
decentrally derived: (1) stabilizing and (2) dissipativating DOF controllers
are shown in Fig. 8. According to these observations (also using the reported
MAO and MAP values in Fig. 8), it is clear that dissipativating controllers,
compared to stabilizing controllers, render smoother and faster output and
performance trajectories in the face of significant disturbances.

##  X Conclusion

Starting from reviewing existing and new LMI-based control solutions for LTI
systems, we presented several decentralized analysis and control synthesis
techniques to verify and ensure properties like stability and dissipativity of
large-scale networked systems. We considered a substantially more general
problem setup than state of the art and developed decentralized processes
covering a broader range of properties of interest. The synthesized control
laws are distributed, and the proposed analysis and control synthesis
processes themselves are decentralized, compositional and resilient to
subsystem removals. We also have shown that optimizing the indexing scheme
used in such distributed processes can substantially reduce the required
information-sharing sessions between subsystems and, in some cases, even make
the overall process distributed. Moreover, we have derived novel centralized
LMI-based solutions for dissipative local observer design and dissipative
local dynamic output feedback controller design problems along with their
decentralized counterparts. Subsequently, we specialized all the derived
results for discrete-time networked systems and provided several simulation
examples to demonstrate the proposed novel decentralized analysis and control
synthesis processes and dissipativity-based results. Future work aims to study
the effect of erroneous and failed information sharing sessions among
subsystems and develop robust distributed analysis and control synthesis
approaches for such scenarios.

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