#  Personality-Driven Gaze Animation with Conditional Generative Adversarial
Networks

Funda Durupinar
Department of Computer Science
University of Massachusetts Boston
Boston, Massachusetts
<EMAIL_ADDRESS>

###### Abstract

We present a generative adversarial learning approach to synthesize gaze
behavior of a given personality. We train the model using an existing data set
that comprises eye-tracking data and personality traits of 42 participants
performing an everyday task. Given the values of Big-Five personality traits
(openness, conscientiousness, extroversion, agreeableness, and neuroticism),
our model generates time series data consisting of gaze target, blinking
times, and pupil dimensions. We use the generated data to synthesize the gaze
motion of virtual agents on a game engine.

_Keywords_ gaze animation, generative adversarial networks, convolutional
neural networks

##  1 Introduction

Expressive eye movements are essential components of believable virtual
character animation. They effectively communicate attention in addition to
giving information about the emotional and mental states of the individual.
There are a number of factors that control and explain the various manners of
gaze behavior, such as turn taking, information processing and scene context.
Personality is one such factor. Studies suggest that there are correlations
between different aspects of personality and gaze parameters such as gaze
shifts and blink rates Libby and Yaklevich (1973); Rauthmann et al. (2012);
Hoppe et al. (2018); Berkovsky et al. (2019).

In this work, we propose a data-driven, generative approach to synthesize gaze
behaviors for different personalities. We use data acquired from individuals
in an everyday setting as opposed to data from actors playing a given role
based on known personality-gaze correlations Ruhland et al. (2015). This helps
capture the small details of gaze cues not yet conceptualized, but reflecting
certain personality traits.

We employ the widely accepted Big-Five model of personality, which describes
personality in five orthogonal dimensions of openness, conscientiousness,
extroversion, agreeableness, and neuroticism Goldberg (1990). We train a
generative adversarial network conditioned on personality classes for each
dimension. Our model learns from an existing personality-annotated dataset by
Hoppe et al. (2018). The dataset includes participants’ Big-Five values and
time-series data for gaze coordinates, blinking times, and pupil dimensions.
We then use the generated data to animate the eye movements of a virtual
model.

Although deep learning has been used to generate gaze movement Klein et al.
(2019), the applications are limited to eye and body pose coordination for
target following. To our knowledge, our method is the first to apply deep
learning to generate eye movement data based on personality expression.

##  2 Related Work

Gaze movement research combines knowledge from various disciplines including
psychology, neuroscience, social sciences, machine learning and computer
graphics. In computer graphics, gaze behavior is animated both by procedural
Lance and Marsella (2008); Peters (2010); Pejsa et al.  and data-driven
approaches Pejsa et al. (2016); Klein et al. (2019).

With the recent advances in eye tracking and machine learning, correlations
between features related to blinks and eyeball movements such as fixations and
saccades with the Big Five personality traits have been established Rauthmann
et al. (2012); Hoppe et al. (2018); Berkovsky et al. (2019). Gaze animation
methods that reflect personality traits apply such correlations to create the
desired effects Ruhland et al. (2015); Fukayama et al. (2002).

Generative Adversarial Networks (GANs) have been highly successful at
synthesizing realistic data. Since their introduction by Goodfellow et al.
Goodfellow et al. (2014), many variations to capture specific conditions have
been proposed Radford et al. (2016); Arjovsky et al. (2017). In addition to
the original image synthesis domain, they have been applied to generating
natural language Yu et al. (2017), time-series data such as health records
Hyland et al. (2017), music Mogren (2016), human motion Barsoum et al. (2018);
Ferstl et al. (2019) and so on. GANs are also popular in computer graphics,
especially for facial animation, locomotion and gesture synthesis Vougioukas
et al. (2018); Sadoughi and Busso (2019); Wang et al. (2019); Ferstl et al.
(2019).

##  3 Method

###  3.1 Data

For training, we use the dataset provided by Hoppe et al. Hoppe et al. (2018).
The dataset consists of binocular eye movement data of 42 participants, each
with an average of 12.51 minutes of tracking information and personality
scores for five factors binned into three groups of low, medium and high. The
data was acquired by head-mounted eye trackers while participants walked
around the campus and purchased an item of their choice from a campus shop.
The data was sampled at 60 Hz. The availability of personality information and
the everyday nature of the performed tasks as opposed to laboratory-confined
tests make this dataset a good fit for our goals.

###  3.2 Gaze Parameter Synthesis

To synthesize gaze parameters, we build a Generative Adversarial Network (GAN)
conditioned on personality values. The GAN is composed of two competing
networks: a discriminator ($D$) and a generator ($G$). G learns a distribution
$p_{g}$ over data $x$ while D is trained to discriminate between the real data
and synthetic data $G(z)$, where $z$ is input noise drawn from a random normal
distribution. The two networks are trained simultaneously. The problem can be
identified as a minimax game with a value function $V(D,G)$ ( Goodfellow et
al. (2014)):

$min_{G}max_{D}V(D,G)=\mathbb{E}_{x\sim p_{r}}[log(D(x))]+\mathbb{E}_{z\sim
p_{g}}[1-log(D(G(z))]$ (1)

Conditional GANs Arnelid et al. (2019); Mirza and Osindero (2014) condition
the model on given classes, allowing direct data generation given class
labels. In our model, we integer encode the personality values into class
labels. The GAN is extended to handle conditional labels $y$ with distribution
$p_{l}$ as:

$min_{G}max_{D}V(D,G)=\mathbb{E}_{x\sim p_{r},y\sim
p_{l}}[log(D(x,y))]+\mathbb{E}_{z\sim p_{g},y\sim p_{l}}[1-log(D(G(z,y),y)]$
(2)

There are different models to make predictions on time series data. For
instance, recurrent neural networks (RNN) perform well to model sequence data,
especially with Long short-term memory (LSTM) units which are effective in
learning longer dependencies. 1D convolutional neural network (CNN) models are
also used for time-series data prediction as they allow time-invariant feature
extraction. They can also be easily extended to multivariate time-series data
as in our case. In our experiments, we found deep CNNs (similar to the DCGAN
architecture by Radford et al. (2016)) to perform better in terms of stability
than RNNs with multiple LSTM units.

The data features include both continuous and discrete parameters. Gaze
coordinates and pupil dimensions are continuous, and they can be directly fed
into the GAN. However, blinking information is categorical. At each time step,
it is zero if no blink was detected, or one if blink was detected. Categorical
data includes an additional step between the generator and the discriminator
to provide continuous gradients. For this, we pre-train an autoencoder to
encode and decode the binary data, and send the output of the generator to the
decoder first, then send the output of the decoder to the discriminator. The
model architecture is shown in Figure 1.

Figure 1: Conditional GAN architecture.

###  3.3 Model Parameters

Discriminator input is a 4-dimensional vector comprising the x and y
coordinates of gaze positions, the average pupil diameter of the left and the
binary blinking data. We organize the data by sliding windows of size 300
corresponding to 5 seconds of data at 60 Hz. For each of these 300-frame
windows, we perform a strict quality test and discard the windows that include
at least one row with x or y coordinates beyond the $[0,1]$ range or pupil
dimensions equal to zero. This leaves only the valid data points. Before
feeding the continuous data into the GAN, we normalize it into the $[-1,1]$
range.

Personality comprises the conditional class label that specifies each
participant. It is introduced to the network as a 50-dimensional embedding
vector that encodes 243 possible values ($3^{5}$ for each bin and personality
dimension). Since the dataset is limited, only 24 out 243 possible classes are
represented in the training set. This conditions the discriminator on the
_seen_ classes, but allows the generator to predict sequences for _unseen_
classes.

We also train the model for each personality dimension separately, where the
personality dimension has three labels representing the low, medium and high
values per personality.

We train mini-batches of size 64, using Adam optimizer with a learning rate of
0.0001 both for D and G. The implementation is done using Keras functional
API.

###  3.4 Evaluation

To evaluate the GAN model quantitatively, we train a deep 1-D CNN classifier
on real data, synthesize a large number of data points and predict the
probability of them belonging to each personality bin (class). Inception score
is a metric to summarize these predictions Salimans et al. (2016). Table 1
shows the scores for the test data for real and synthetic values. When we
train the classifier on all the five dimensions, because the representation in
the training data is limited, the inception score is low. We also compute the
inception scores when each dimension is introduced as a condition separately.
Considering that there are three classes per personality dimension, the closer
the inception score to 3, the better the results. We see that extroversion
performs the best and neuroticism the worst.

Table 1: Inception scores per personality for synthetic and real data Data | O | C | E | A | N | All dims
---|---|---|---|---|---|---
Synthetic | 2.38 | 2.25 | 2.56 | 2.41 | 2.56 | 6.23
Real test | 2.87 | 2.63 | 2.88 | 2.78 | 2.89 | 15.62

To analyze the synthetic gaze trajectory data visually, we plot the average
trajectory for each personality dimension and class for 1000 generated data
points spanning 300 time steps (5 seconds). For each time step in the 5-second
window, we compute the average x and y values. Figure 2 shows that gaze
coordinates for each personality class are clearly distinguishable from each
other. These trajectories are consistent with some of the findings by Fukuyama
et al. Fukayama et al. (2002). They report that looking up is rated as being
dominant, a trait of extroversion. They find looking down to be associated
with anxiety, which confirms the decreasing values for our neuroticism data.
Of course, perception of personality and its actual expression are different
concepts, and not every cue is an indicator of personality. However, such
clear associations lead the way for future research directions.

Figure 2: Synthesized average x and y coordinates for each OCEAN dimension and
class: blue for low, green for medium, and red for high values.

We also visualize the average pupil size for each personality dimension and
class for each time step in Figure 3. Overall, larger pupil sizes seem to be
associated with positive traits (considering emotional stability as the
positive pole).

Figure 3: Synthesized pupil sizes for each OCEAN dimension and class: blue for
low, green for medium, and red for high values.

Possibly due to the discrete nature of blinking features, the generated data
for all the personality types converge to the same result.

###  3.5 Gaze Animation

To animate eye movement, pupil dilation and blinking with respect to the data
we created 3D human models by Adobe Fuse Adobe (2019). The model includes
blend shapes for face, making it easy to animate facial expressions, and in
our case, blinking motion. Figure 4 shows the same humanoid model animated
with gaze motions of different personalities.

Figure 4: Animated gaze of an extrovert (left) vs. an introvert (right) model.

The eye tracker glasses that were used to capture gaze data (SMI) have $None$
horizontal and $None$ vertical field of view angles. Using these angles, we
convert the x and y values which are in the range $[0,1]$ corresponding to the
device space coordinates to the world space. The target in the world space is
the look-at direction of the eyes. For convenience, we take the middle point
of the left and right eyes as the eye position. The target position is
computed as follows:

$\displaystyle target_{world_{x}}$ $\displaystyle=$
$\displaystyle(2x-1)d|tan($$)|+eye_{x}$ $\displaystyle target_{world_{y}}$
$\displaystyle=$ $\displaystyle(2y-1)d|tan($$)|+eye_{y}$ $\displaystyle
target_{world_{z}}$ $\displaystyle=$ $\displaystyle d+eye_{z}$

, where $d$ is the maximum viewing distance. Note that the viewing direction
from the eye position to the target point will be along the same line
regardless of the viewing distance $d$, so this value will be canceled out
later when computing the eyeball transforms. In addition to rotating the
eyeballs to align with the look-at vector, we update the weights of the eyelid
blendshapes so that they move naturally when the eyes move up and down.

We use a separate eyeball mesh to simulate the pupil dilation. We update the
pupil dimension by applying forces to the vertices on the pupil perimeter
towards or out of the center of the pupil as shown in Figure 5.

Figure 5: Animating pupil dilation.

##  4 Conclusion

We introduce a generative deep learning approach to synthesize time-series
gaze data, and animate it on a virtual character. Our method is a preliminary
step in this direction. The next step will be to create a personality-
annotated gaze dataset during a conversation and to use our generative
approach to this data. The data will include similar features, but
additionally, it will have information about the conversation target as well
as head and torso pose information. We believe that the social nature of the
task will help capture more salient features of personality expression.

## References

  * Libby and Yaklevich [1973] W. L. Libby and D. Yaklevich.  Personality determinants of eye contact and direction of gaze aversion.  _Journal of Personality and Social Psychology_ , 24(2):197–206, 1973.
  * Rauthmann et al. [2012] John F. Rauthmann, Christian Seubert, Pierre Sachse, and Marco Furtner.  Eyes as windows to the soul: Gazing behavior is related to personality.  _Journal of Research in Personality_ , 46:147–156, 2012\.
  * Hoppe et al. [2018] Sabrina Hoppe, Tobias Loetscher, Stephanie A. Morey, and Andreas Bulling.  Eye movements during everyday behavior predict personality traits.  _Frontiers in Human Neuroscience_ , 12:105, 2018.  ISSN 1662-5161.  doi:10.3389/fnhum.2018.00105.  URL https://www.frontiersin.org/article/10.3389/fnhum.2018.00105.
  * Berkovsky et al. [2019] Shlomo Berkovsky, Ronnie Taib, Irena Koprinska, Eileen Wang, Yucheng Zeng, Jingjie Li, and Sabina Kleitman.  Detecting personality traits using eye-tracking data.  In _Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems_ , CHI ’19, page 1–12, New York, NY, USA, 2019. Association for Computing Machinery.  ISBN 9781450359702.  doi:10.1145/3290605.3300451.  URL https://doi.org/10.1145/3290605.3300451.
  * Ruhland et al. [2015] Kerstin Ruhland, Katja Zibrek, and Rachel McDonnell.  Perception of personality through eye gaze of realistic and cartoon models.  In _Proceedings of the ACM SIGGRAPH Symposium on Applied Perception - SAP’15_. ACM Press, 2015.  doi:10.1145/2804408.2804424.
  * Goldberg [1990] Lewis R. Goldberg.  An alternative "description of personality": The big-five factor structure.  _Journal of Personality and Social Psychology_ , 59:1216–1229, 1990.
  * Klein et al. [2019] Alex Klein, Zerrin Yumak, Arjen Beij, and A. Frank van der Stappen.  Data-driven gaze animation using recurrent neural networks.  In _Motion, Interaction and Games_ , MIG ’19. Association for Computing Machinery, 2019.  ISBN 9781450369947.
  * Lance and Marsella [2008] Brent J. Lance and Stacy C. Marsella.  A model of gaze for the purpose of emotional expression in virtual embodied agents.  In _Proceedings of the 7th International Joint Conference on Autonomous Agents and Multiagent Systems - Volume 1_ , AAMAS ’08, page 199–206, Richland, SC, 2008. International Foundation for Autonomous Agents and Multiagent Systems.  ISBN 9780981738109.
  * Peters [2010] C. Peters.  Animating gaze shifts for virtual characters based on head movement propensity.  In _2010 Second International Conference on Games and Virtual Worlds for Serious Applications_ , pages 11–18, 2010.
  * [10] Tomislav Pejsa, Bilge Mutlu, and Michael Gleicher.  Stylized and performative gaze for character animation.  _Computer Graphics Forum_ , 32(2pt2):143–152.
  * Pejsa et al. [2016] Tomislav Pejsa, Daniel Rakita, Bilge Mutlu, and Michael Gleicher.  Authoring directed gaze for full-body motion capture.  _ACM Trans. Graph._ , 35(6), November 2016.
  * Fukayama et al. [2002] Atsushi Fukayama, Takehiko Ohno, Naoki Mukawa, Minako Sawaki, and Norihiro Hagita.  Messages embedded in gaze of interface agents — impression management with agent’s gaze.  In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ , CHI ’02, page 41–48, New York, NY, USA, 2002. Association for Computing Machinery.  ISBN 1581134533.
  * Goodfellow et al. [2014] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.  Generative adversarial nets.  In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, editors, _Advances in Neural Information Processing Systems 27_ , pages 2672–2680. Curran Associates, Inc., 2014.  URL http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf.
  * Radford et al. [2016] A. Radford, Luke Metz, and Soumith Chintala.  Unsupervised representation learning with deep convolutional generative adversarial networks.  _CoRR_ , abs/1511.06434, 2016.
  * Arjovsky et al. [2017] Martin Arjovsky, Soumith Chintala, and Léon Bottou.  Wasserstein generative adversarial networks.  In _Proceedings of the 34th International Conference on Machine Learning - Volume 70_ , ICML’17, page 214–223. JMLR.org, 2017.
  * Yu et al. [2017] Lantao Yu, Weinan Zhang, Jun Wang, and Yong Yu.  Seqgan: Sequence generative adversarial nets with policy gradient.  In _Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence_ , AAAI’17, page 2852–2858. AAAI Press, 2017.
  * Hyland et al. [2017] Stephanie L. Hyland, Cristóbal Esteban, and Gunnar Rätschm.  Real-valued (medical) time series generation with recurrent conditional gans, 2017.
  * Mogren [2016] Olof Mogren.  C-rnn-gan: Continuous recurrent neural networks with adversarial training, 2016.
  * Barsoum et al. [2018] Emad Barsoum, John Kender, and Zicheng Liu.  Hp-gan: Probabilistic 3d human motion prediction via gan.  _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)_ , pages 1499–149909, 2018.
  * Ferstl et al. [2019] Ylva Ferstl, Michael Neff, and Rachel McDonnell.  Multi-objective adversarial gesture generation.  In _Motion, Interaction and Games_ , New York, NY, USA, 2019. Association for Computing Machinery.  ISBN 9781450369947.  doi:10.1145/3359566.3360053.  URL https://doi.org/10.1145/3359566.3360053.
  * Vougioukas et al. [2018] Konstantinos Vougioukas, S. Petridis, and M. Pantic.  End-to-end speech-driven facial animation with temporal gans.  In _BMVC_ , 2018.
  * Sadoughi and Busso [2019] N. Sadoughi and C. Busso.  Speech-driven expressive talking lips with conditional sequential generative adversarial networks.  _IEEE Transactions on Affective Computing_ , pages 1–1, 2019.
  * Wang et al. [2019] Z. Wang, J. Chai, and S. Xia.  Combining recurrent neural networks and adversarial training for human motion synthesis and control.  _IEEE Transactions on Visualization and Computer Graphics_ , pages 1–1, 2019.
  * Arnelid et al. [2019] H. Arnelid, E. L. Zec, and N. Mohammadiha.  Recurrent conditional generative adversarial networks for autonomous driving sensor modelling.  In _2019 IEEE Intelligent Transportation Systems Conference (ITSC)_ , pages 1613–1618, 2019.
  * Mirza and Osindero [2014] M. Mirza and Simon Osindero.  Conditional generative adversarial nets.  _ArXiv_ , abs/1411.1784, 2014.
  * Salimans et al. [2016] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen.  Improved techniques for training gans, 2016.
  * Adobe [2019] Adobe.  Fuse CC.  https://www.adobe.com/tr/products/fuse.html, 2019.  Accessed: 2019-08-14.


# Extended Full Block S-Procedure for Distributed Control of Interconnected
Systems*

Giulia De Pasquale1, Yvonne R. Stürz2, Maria Elena Valcher1 and Roy S. Smith3
1The authors are with the Dipartimento di Ingegneria dell’Informazione
Università di Padova, Via Gradenigo 6b, 35131 Padova, Italy
<EMAIL_ADDRESS>2The author is with the Model Predictive
Control Laboratory, University of California, Berkeley, CA 94709, USA. Email
address<EMAIL_ADDRESS>3The author is with the Automatic Control
Laboratory, ETH Zurich, Physikstrasse 3, 8092 Zurich, Switzerland
<EMAIL_ADDRESS>project has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No. 846421. Paper accepted at the 59th IEEE
Conference on Decision and Control (CDC), Jeju Island, Korea.

###### Abstract

This paper proposes a novel method for distributed controller synthesis of
homogeneous interconnected systems consisting of identical subsystems. The
objective of the designed controller is to minimize the $\mathcal{L}_{2}$-gain
of the performance channel. The proposed method is an extended formulation of
the Full Block S-Procedure (FBSP) where we introduce an additional set of
variables. This allows to relax the block-diagonal structural assumptions on
the Lyapunov and multiplier matrices required for distributed control design,
which reduces conservatism w.r.t. most existing approaches. We show how to
decompose the proposed extended FBSP into small synthesis conditions, of the
size of one individual subsystem.

##  I Introduction

A wide variety of modern control applications involve complex systems that are
characterized by their large scale, their distributed nature and the sparse
structure of their physical interconnections [1]. Examples include smart
grids, the internet, automated highways [1], satellite formation flying [2],
car platoons [3], cooperative manipulation [4], unmanned aerial vehicles [5]
and large telescopes [6]. This gave rise to an increased interest in the
research on modelling and control of complex distributed systems, consisting
of many subsystems interacting through a network. Distributed systems
represent a class of interconnected systems. When systems are physically
interconnected we refer to them as ”spatially distributed systems” and this is
the case of automated highway systems [7], airplane formation flight,
satellite constellations [2]. We call, instead, ”virtually interconnected
systems” those systems whose subsystems are not spatially interconnected, but
share information in order to reach a common goal, such as mobile robots [8].

The control of large-scale networked systems is a challenging problem since a
large number of inputs and outputs are involved. Because of limitations in
computation and communication, the synthesis and the implementation of a
centralized controller becomes intractable in practice, and decentralized or
distributed control architectures are required. However, in applications with
strong interactions among the subsystems, a completely decentralized
architecture of the controller may not guarantee good performance [9]. With a
growing number of subsystems, the controller synthesis becomes more complex.
Therefore, decomposition methods have been proposed for a scalable controller
synthesis. To this end, the system can be modeled as consisting of a
decentralized part and an interconnection part. This is achieved by
introducing an interconnection channel, that captures the interconnections
among subsystems, and by performing a linear fractional transformation (LFT)
on the system model [10]. Groups of identical interconnected subsystems
constitute a homogeneous system [6]. This model has been extended to groups of
homogeneous systems [11], and groups of homogeneous interconnections [12, 13].
For groups of homogeneous systems, the system structure can be exploited in
order to derive a compact controller synthesis. In [6, 11, 13], the Full Block
S-Procedure (FBSP) has been exploited for the synthesis of a distributed
controller, resulting in linear matrix inequalities (LMIs). The decomposition
and thus scalability of the synthesis conditions are achieved by a block-
diagonal structure of the Lyapunov and multiplier matrices, which introduces
conservatism. In this paper, we propose a novel distributed controller
synthesis for linear time invariant homogeneous interconnected systems, which
allows to reduce this conservatism. This is achieved by proposing an extended
version of the FBSP which involves an additional set of variables. This allows
us to relax the imposed structure on the Lyapunov and multiplier matrices, and
can thus lead to a better control performance. A similar approach has been
proposed in [14] with an extended version of the Bounded Real Lemma. The
extended FBSP (EFBSP) proposed in this paper can be decomposed to allow for a
scalable distributed controller synthesis. We show that by decomposing the
EFBSP we achieve the same degree of scalability w.r.t. the FBSP with block-
diagonal structured Lyapunov and multiplier matrices, namely a decomposition
into small matrix inequalities which are of the size of the individual
subsystems is achieved.

The paper is structured as follows. In Section II interconnected systems are
introduced with special focus on homogeneous systems. Section III formalizes
the synthesis problem of the interconnected controller. In Sections IV-A and
IV-B the primal and dual formulations of the EFBSP are given, while in Section
V the decomposition of the controller synthesis equations based on the EFBSP
is introduced. Section VI shows some numerical results.

Notation We denote a block-diagonal matrix $D$ with diagonal blocks
$D_{1},...,D_{N}$ by
$D=\mathrm{\operatorname*{diag}}_{i=1}^{N}\\!\left(D_{h}\right)$. The $n\times
n$-identity matrix is denoted by $I_{n}$ and the $n\times m$ matrix of all
zeros by $0_{n\times m}$. If clear from the context, the indices are dropped.
Given a matrix $M$, its minimum and maximum singular values are denoted by
$\sigma_{\mathrm{min}}(\cdot)$ and $\sigma_{\mathrm{max}}(\cdot)$,
respectively, while $spec(M)$ denotes its spectrum, i.e. the set of all its
eigenvalues. The Kronecker product is denoted by ${}\otimes{}$. Given a
complex valued matrix
$M=\small\left[\begin{array}[]{@{\,}c@{\,\;}c@{\,}}M_{1}&M_{2}\\\
M_{3}&M_{4}\end{array}\right]$ and a matrix $P$ of appropriate dimensions, the
lower and upper Linear Fractional Transformation (LFT) associated with the
pair $(M,P)$ are defined as
$\mathcal{F}_{\ell}(M,P)=M_{1}+M_{2}P(I-M_{4})^{-1}M_{3}$ and
$\mathcal{F}_{\text{u}}(M,P)=M_{4}+M_{3}P(I-M_{1})^{-1}M_{2}$, respectively.
We use the symbol $\star$ to simplify expressions as $M_{1}^{\top}M_{2}M_{1}$,
i.e., $\star^{\top}M_{2}M_{1}=M_{1}^{\top}M_{2}M_{1}$.

##  II Homogeneous Interconnected Systems

In this paper, we consider homogeneous interconnected systems which are
composed of $N$ identical subsystems. The subsystems can be coupled through
their dynamics, disturbance, performance goal or communication. Assuming that
each subsystem, denoted by $G_{h}$, $h\in\mathcal{N}$,
$\mathcal{N}:=\\{1,\dots,N\\}$, is of order $n_{h}$$=n$, the state-space
dynamics of the whole system, of order $n_{x}:=\sum_{h\in\mathcal{N}}n_{h}$ =
$Nn$, is described as

$\begin{cases}\dot{x}(t)&=Ax(t)+B_{w}w(t)+B_{u}u(t)\\\
y(t)&=C_{y}x(t)+D_{yw}w(t)\\\
z(t)&=C_{z}x(t)+D_{zw}w(t)+D_{zu}u(t)\end{cases}$ (1)

with the state vector $x\in\mathbb{R}^{n_{x}}$, the control input and measured
output, $u\in\mathbb{R}^{n_{u}}$ and $y\in\mathbb{R}^{n_{y}}$, the exogenous
input and performance output, ${w}\in\mathbb{R}^{n_{{w}}}$ and
${z}\in\mathbb{R}^{n_{{z}}}$.

We define the undirected and unweighted graph that represents the coupling
among the subsystems (in terms of dynamics, performance or disturbance) as
$\mathcal{G}=\\{\mathcal{N},\mathcal{E}\\}$, where the index set $\mathcal{N}$
represents the $N$ subsystems $G_{h}$, and
$\mathcal{E}\subseteq\mathcal{N}\times\mathcal{N}$ represents the
interconnections between the subsystems. In particular, the pair $(h,j)$
belongs to $\mathcal{E}$, for $h\neq j$, if and only if subsystems $G_{j}$ and
$G_{h}$ communicate. The interconnections among the subsystems $G_{h}$ are
described by a pattern matrix denoted by $P$: its $(h,j)$th entry is $1$ if
and only if $(h,j)\in\mathcal{E}$, otherwise it is zero. We assume that
${\mathcal{G}}$ has no self loops and hence the diagonal entries of $P$ are
zero. The pattern matrix $P$ fully characterizes the topology of the
interactions among the subsystems. For simplicity, we have considered
undirected graphs, i.e., we have assumed that the pattern matrix $P$ is
symmetric. This ensures that the eigenvalues of $P$ are real. However, the
present results can be extended to directed graphs as well, as discussed in
[15].

###  II-A Homogeneous Interconnected State-Space Representation

For a homogeneous interconnected system, the system matrices in (1) have the
following structure:

######  Definition 1 (Homogeneous Decomposable System)

Let $M$ generically represent one of the system matrices $A$, $B_{w}$,
$B_{u}$, $C_{y}$, $D_{y{w}}$, $C_{{z}}$, $D_{{z}u}$, $D_{{z}{w}}$, in (1).
System (1) is said to be a homogeneous decomposable system if each of its
system matrices can be written as

$M={I_{N}}\otimes{{M^{d}}}+{{P}}\otimes{M^{i}},$ (2)

with ${P}$ being the pattern matrix defined before, for two suitable matrices
$M^{d}$ and $M^{i}$. This means that for a homogeneous system all local
subsystem matrices (on the diagonal) are identical and equal to $M^{d}$, and
all interconnection subsystem matrices are equal to $M^{i}$ (the off-diagonal
block $(h,k),h\neq k,$ is equal to $M^{i}$ if $(h,k)\in{\mathcal{E}}$, the
zero matrix otherwise).

Adopting an approach similar to [11], the system can be modeled through an LFT
of a decentralized part and an interconnection part, as a result of the
introduction of an interconnection channel. The diagonal-blocks of the system
matrices (1) model the decentralized part and the off-diagonal ones refer to
the interconnection part. We introduce the interconnection channel
${q\rightarrow p}$ as

${p=\mathcal{P}q},$ (3)

where  $p\in\mathbb{R}^{n_{p}}$ and $q\in\mathbb{R}^{n_{q}}$ are the
interconnection variables. Specifically, the $h$-th block of $p$ represents
the interconnection variables available as inputs to the $h$-th subsystem,
while the $j$-th block of $q$ represents the variables of subsystem $G_{j}$
that are accessible to those subsystems $G_{h}$ for which
$(h,j)\in\mathcal{E}$. The interconnection matrix $\mathcal{P}$ in (3)
captures the interconnection relations, hence the system can be modeled by the
LFT $\mathcal{F}_{\ell}(G,\mathcal{P})$, with $G=I_{N}\otimes G_{h}$, where
$G_{h}$ are the decentralized parts of the subsystems. For ideal deterministic
interconnections considered in this work $\mathcal{P}$ can be defined as
$\mathcal{P}={P}\otimes{I_{n_{q}}}$.

The system vector in (1) is structured as
$x=[x_{1}^{\top},...,x_{N}^{\top}]^{\top}$, and analogously the input and
output vectors, and the interconnection variables in (3). The $N$ subsystems
$G_{h}$ of the system in (1) then admit a continuous-time state-space
representations given by

$\displaystyle G_{h}$
$\displaystyle:\begin{cases}\left[\begin{array}[]{@{}c@{}}\dot{x}_{h}\\\
\hdashline y_{h}\\\ \hdashline{z}_{h}\\\ \hdashline q_{h}\\\
\end{array}\right]=\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,} :
c@{\,\,\,\,\,\,} : c@{\,\,\,\,\,\,} :
c@{}}{A}_{h}&{B}_{u_{h}}&{B}_{{w}_{h}}&{B}_{p_{h}}\\\
\hdashline{C}_{y_{h}}&{D}_{yu_{h}}&{D}_{y{w}_{h}}&{D}_{yp_{h}}\\\
\hdashline{C}_{{z}_{h}}&{D}_{{z}u_{h}}&0&{D}_{{z}p_{h}}\\\
\hdashline{C}_{q_{h}}&{D}_{qu_{h}}&{D}_{q{w}_{h}}&0\\\
\end{array}\right]\left[\begin{array}[]{@{}c@{}}x_{h}\\\ \hdashline u_{h}\\\
\hdashline{w}_{h}\\\ \hdashline p_{h}\end{array}\right],\hskip
2.5pth\in\mathcal{N},\end{cases}$ (4)

with the state vector $x_{h}\in\mathbb{R}^{n_{x_{h}}}$, the local control
input and measublack output, ${u}_{h}\in\mathbb{R}^{n_{u_{h}}}$ and
${y}_{h}\in\mathbb{R}^{n_{y_{h}}}$, the local exogenous input and performance
output, ${w}_{h}\in\mathbb{R}^{n_{{w}_{h}}}$ and
${z}_{h}\in\mathbb{R}^{n_{{z}_{h}}}$, and the interconnection signals
$p_{h}\in\mathbb{R}^{n_{p_{h}}}$ and $q_{h}\in\mathbb{R}^{n_{q_{h}}}$,
respectively111 It is worth noticing that modeling the system with the
interconnection channel $p$ to $q$ is equivalent to the system model in (1)..
In this paper we assume $q_{h}=[{x_{h}}^{\top},{u_{h}}^{\top}]^{\top}$, namely
that $C_{q_{h}}=\begin{bmatrix}I\\\ 0\end{bmatrix}$,
$D_{qu_{h}}=\begin{bmatrix}0&0\\\ 0&I\end{bmatrix}$ and $D_{qw_{h}}=0$.
Consequently, $p_{h}=[{x_{h}^{i}}^{\top},{u_{h}^{i}}^{\top}]^{\top}$, where
${x_{h}^{i}}^{\top},{u_{h}^{i}}^{\top}$ are the overall interconnected state
and control action of the neighboring subsystems $G_{j}$ of subsystem $G_{h}$
222In other words, $x_{h}^{i}=\sum_{j:[P]_{h,j}=1}x_{j}$ and
$u_{h}^{i}=\sum_{j:[P]_{h,j}=1}u_{j}$, according to (3). .

Based on Definition 1 and equation (2), the state-space matrices
$A_{h},B_{uh},C_{yh},B_{wh},D_{ywh},C_{zh},D_{zuh}$ in (4) are the block-
diagonal parts of the system matrices in (1), and the system matrices
belonging to the interconnection channel
$C_{qh},D_{quh},D_{qwh},B_{ph},D_{yph},D_{zph}$ are composed of the off-block-
diagonal subsystem matrices from (1), i.e.,

$\displaystyle B_{p_{h}}$
$\displaystyle=[{A^{i}},B^{i}_{u}],D_{{yp}_{h}}=[{C}^{i}_{y},{D}^{i}_{{yu}}],D_{{{z}p}_{h}}=[{C}^{i}_{{z}},{D}^{i}_{{{z}u}}],$
(5) $\displaystyle C_{q_{h}}$
$\displaystyle=C^{d}_{q_{h}},D_{qu_{h}}=[{D}^{d}_{{qu}},{D}^{i}_{{qu}}],D_{qw_{h}}={D}^{d}_{{q{w}}}.$

Note that the representations in (5) is not unique: choosing the
interconnection channel is a modeling choice.

###### Remark 1

We will assume $D^{d}_{zw}$ to be zero since the feedthrough matrix
$D^{d}_{zw}$ is often chosen to be zero by convention.

##  III Interconnected Controller Synthesis

We consider interconnected static state feedback controllers, whose
interconnection topology is the same as the one of the system, i.e., the
pattern matrix of the controller is equal to the one of the system. The
controller $K$ for the distributed state feedback $u(t)=Kx(t),$ can thus be
written as

$K={I_{N}}\otimes{K^{d}}+{P}\otimes{K^{i}},$ (6)

with $K^{d}$ being the decentralized local controller gain and $K^{i}$ the
interconnected controller gain.

We will present a method to synthesize a controller as in (6) for homogeneous
systems described as in Definition 1, that minimizes the
$\mathcal{H}_{\infty}$ norm of the transfer function of the closed-loop
system, from the exogenous input $w$ to the performance output $z$, denoted by
$T_{zw}$.

The advantage of designing a distributed controller with the structure given
in (6) lies in the fact that the closed-loop system matrices have the same
structure as in (2), which will prove to be advantageous in terms of
scalability of the controller synthesis equations.  If we apply such a state
feedback law to the system in (4) in closed-loop, we obtain:

$\small\left[\begin{array}[]{@{}c@{}}\dot{x}\\\ q\\\
z\end{array}\right]=\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}c@{}}\mathcal{A}&\mathcal{B}_{1}&\mathcal{B}_{2}\\\
\mathcal{C}_{1}&\mathcal{D}_{11}&\mathcal{D}_{12}\\\
\mathcal{C}_{2}&\mathcal{D}_{21}&\mathcal{D}_{22}\end{array}\right]\\!\\!\\!\\!\small\left[\begin{array}[]{@{}c@{}}x\\\
p\\\ w\end{array}\right],\quad p=\mathcal{P}q,$ (7)

where the meanings and sizes of the variables $x$, $q$ and $z$ are the same as
in the previous section and

$\displaystyle\mathcal{A}={I_{N}}\otimes{(A^{d}+[B^{d}_{u},B^{i}_{u}]\hat{K})},\quad\mathcal{B}_{1}={I_{N}}\otimes{[A^{i},B^{d}_{u}]},$
$\displaystyle\mathcal{B}_{2}={I_{N}}\otimes{B^{d}_{{{w}}}},\quad\mathcal{C}_{1}={I_{N}}\otimes{(C^{d}_{{q}}+[D^{d}_{{qu}},D^{i}_{{qu}}]\hat{K})},$
$\displaystyle\mathcal{D}_{11}=0,\quad\mathcal{D}_{12}={I_{N}}\otimes{D^{d}_{{q{w}}}}$
(8)
$\displaystyle\mathcal{C}_{2}={I_{N}}\otimes{(C^{d}_{{z}}+[D^{d}_{{{z}u}},D^{i}_{{{z}u}}]\hat{K})},$
$\displaystyle\mathcal{D}_{21}={I_{N}}\otimes{[C^{i}_{{z}},D^{d}_{{zu}}]},\quad\mathcal{D}_{22}=0,$

with $\hat{K}=\begin{bmatrix}K^{d\top},K^{i\top}\end{bmatrix}^{\top}$. In this
paper we will assume the system (1) to be stabilizable and the system (7) to
be well-posed, see [16]. In the following, we present a novel method to
synthesize the controller, which is based on the FBSP presented in [16]. It is
an extended formulation in which an additional set of variables is introduced
according to the Extended Bounded Real Lemma (EBRL), presented in [14]. This
allows us to decouple the Lyapunov matrix from the other optimization matrices
in the EFBSP, thus relaxing the block-diagonal structure of the Lyapunov
matrix. This will prove to reduce the conservatism with respect to the
standard FBSP used in [6, 13], where block-diagonal Lyapunov and multiplier
matrices are assumed.

##  IV Extended Full Block S-Procedure

###  IV-A Extended Full Block S-Procedure in Primal Form

In this section we propose a novel method for the synthesis of a distributed
state-feedback controller for the continuous time closed loop system described
as in (7) and (III). The proposed method is an extension of the state-feedback
synthesis problem proposed in [16], the FBSP that we recall in the following
theorem.

###### Theorem 1

Full Block S-Procedure [16] Given an asymptotically stable continuous-time LTI
system described as in (7), then the system has an $\mathcal{L}_{2}$-gain from
$w$ to $z$ smaller than $\gamma$ if and only if there exist matrices
$\mathcal{X}=\mathcal{X}^{\top}>0$, $R=R^{\top}$, $Q=Q^{\top}$ and $S$ of
appropriate dimensions such that

$\displaystyle\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{T}\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,}
c@{}}Q&S\\\
S^{\top}&R\end{array}\right]^{\top}\small\left[\begin{array}[]{@{}c@{}}\mathcal{P}\\\
I\end{array}\right]$ $\displaystyle>0,$ (14)
$\displaystyle\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{\top}\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,}
c@{\,\,\,\,\,} : c@{\,} c@{\,} : c@{\,\,\,\,\,}
c@{\,\,\,\,\,}}0&\mathcal{X}&0&0&0&0\\\ \mathcal{X}&0&0&0&0&0\\\ \hdashline
0&0&Q&S&0&0\\\ 0&0&S^{\top}&R&0&0\\\ \hdashline 0&0&0&0&-\gamma^{2}I&0\\\
0&0&0&0&0&I\end{array}\right]\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}
c@{\,\,\,\,\,\,} c@{}}I&0&0\\\ \mathcal{A}&\mathcal{B}_{1}&\mathcal{B}_{2}\\\
\hdashline 0&I&0\\\ \mathcal{C}_{1}&\mathcal{D}_{11}&\mathcal{D}_{12}\\\
\hdashline 0&0&I\\\
\mathcal{C}_{2}&\mathcal{D}_{21}&\mathcal{D}_{22}\end{array}\right]$
$\displaystyle<0.$ (28)

Equations (14) and (28) are known in the literature as multiplier condition
and nominal condition, respectively. In this work, the FBSP will be exploited
for the design of distributed controllers [6]. A decomposed scalable
distributed control design can be obtained by imposing on the Lyapunov matrix
$X$ and the multiplier matrices $Q,R,S$ a block-diagonal structure, which in
general introduces conservatism. In order to reduce this conservatism, we
propose an extended formulation of the FBSP applied to distributed control
presented in [6] which we refer to as EFBSP. We first present the primal
formulation, and then dualize it for implementation purposes. The primal
formulation is given in the following theorem.

###### Theorem 2

Extended FBSP in Primal Form The well-posed rational interconnected system in
(7) has the $\mathcal{L}_{2}$-gain from $w$ to $z$ smaller than $\gamma$ if
and only if there exist matrices $X=X^{\top}>0$, $F$ non-singular, and
$Q=Q^{\top}$, $R=R^{\top}$, $S$, such that

$\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{\top}\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}c@{}}Q&S\\\
S^{\top}&R\end{array}\right]\left[\begin{array}[]{@{}c@{}}\mathcal{P}\\\
I\end{array}\right]>0,$ (29)
$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{\top}\\!\\!\\!\footnotesize\left[\begin{array}[]{@{}c@{\,}
c@{\\!} @{\,} c @{\\!\\!\\!\\!\\!\\!\\!\\!} @{\,\,\,\,\,\,\,\,\,\,\,\,}c
@{\\!\\!\\!\\!\\!\\!\\!\\!}: @{\,\,\,\,\,\,\,\,\,\,\,\,\,\,}c@{\,\,\,\,\,\,}
c@{\,\,\,\,\,\,} : @{\,\,}c@{\,\,\,\,\,\,}c@{}}0&X-F&F&0&0&0&0&0\\\
X-F^{\top}&-F-F^{\top}&0&F&0&0&0&0\\\ F^{\top}&0&0&0&0&0&0&0\\\
0&F^{\top}&0&0&0&0&0&0\\\ \hdashline 0&0&0&0&Q&S&0&0\\\
0&0&0&0&S^{\top}&R&0&0\\\ \hdashline 0&0&0&0&0&0&-{I\gamma^{2}}&0\\\
0&0&0&0&0&0&0&I\end{array}\right]\\!\\!\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}
: c@{} :c@{} }I&0&0&0\\\ 0&I&0&0\\\
\mathcal{A}&0&\mathcal{B}_{1}&\mathcal{B}_{2}\\\
\mathcal{A}&0&\mathcal{B}_{1}&\mathcal{B}_{2}\\\ \hdashline 0&0&I&0\\\
\mathcal{C}_{1}&0&\mathcal{D}_{11}&\mathcal{D}_{12}\\\ \hdashline 0&0&0&I\\\
\mathcal{C}_{2}&0&\mathcal{D}_{21}&\mathcal{D}_{22}\end{array}\right]\\!\\!<\\!0$
(30)

  * Proof:

Consider the extended robust performance conditions expressed in Theorem 2 of
[14]. We note that these conditions can equivalently be expressed as follows:
There exist $X=X^{\top}>0$ and $F$ non-singular, such that

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{\top}\footnotesize\left[\begin{array}[]{@{}c@{\,}
c@{\\!} @{\,} c @{\\!\\!\\!\\!\\!\\!\\!\\!\\!}
@{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!} c@{\,} :
@{\,\,\,\,\,\,\,\,\,\,\,\,\,} c @{\,\,\,\,\,\,\,}c@{}}0&X-F&0&F&0&0\\\
X-F^{\top}&-F-F^{\top}&F&0&0&0\\\ F^{\top}&0&0&0&0&0\\\ 0&F^{\top}&0&0&0&0\\\
\hdashline 0&0&0&0&-{\gamma^{2}I}&0\\\
0&0&0&0&0&I\end{array}\right]\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}:c@{}}I&0&0\\\
0&I&0\\\ \mathcal{A}(\mathcal{P})&0&\mathcal{B}_{1}(\mathcal{P})\\\
\mathcal{A}(\mathcal{P})&0&\mathcal{B}_{1}(\mathcal{P})\\\ \hdashline 0&0&I\\\
\mathcal{C}_{1}(\mathcal{P})&0&\mathcal{D}_{11}(\mathcal{P})\end{array}\right]<0,$
(31)

where the matrices $\mathcal{A}(\mathcal{P})$, $\mathcal{B}_{1}(\mathcal{P})$,
$\mathcal{C}_{1}(\mathcal{P})$ and $\mathcal{D}_{11}(\mathcal{P})$ are the
closed-loop system matrices of

$\begin{cases}\dot{x}(t)&=\mathcal{A}(\mathcal{P})x(t)+\mathcal{B}_{1}(\mathcal{P})w(t)\\\
z(t)&=\mathcal{C}_{1}(\mathcal{P})x(t)+\mathcal{D}_{11}(\mathcal{P})w(t),\end{cases}$
(32)

defined as
$\mathcal{A}(\mathcal{P})={I_{N}}\otimes{(A^{d}+B^{d}_{u}K^{d})}+{P}\otimes{(A^{i}+B^{d}_{u}K^{i})}$,
and the other state-space matrices are defined in the same way, in accordance
with (2), $\mathcal{B}_{1}(\mathcal{P})={I_{N}}\otimes{B^{d}_{w}}$,
$\mathcal{C}_{1}(\mathcal{P})={I_{N}}\otimes{(C^{d}+D^{d}_{u}K^{d})}+{P}\otimes{(C^{i}+D^{d}_{u}K^{i})}$,
$\mathcal{D}_{11}(\mathcal{P})={I_{N}}\otimes{D^{d}_{w}}$.

Now we prove the equivalence between (31) and (29)- (30). To do so, we make
use of the FBSP in Theorem 5 in the Appendix. Equation (31), with the
following definitions

$\displaystyle x$ $\displaystyle=\small\left[\begin{array}[]{@{}c@{}}x\\\
\dot{x}\\\ \dot{x}\\\ \dot{x}\\\ p\\\ q\\\ w\\\ z\end{array}\right],\quad
W=\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}:c@{\,\,\,\,\,\,}c@{}}0&X-F&F&0&0&0\\\
X-F&-F-F^{\top}&0&F&0&0\\\ F^{\top}&0&0&0&0&0\\\ 0&F^{\top}&0&0&0&0\\\
\hdashline 0&0&0&0&-\gamma^{2}I&0\\\ 0&0&0&0&0&I\end{array}\right],$ (33)
$\displaystyle V$
$\displaystyle=\small\left[\begin{array}[]{@{}c@{\,\,\,\,}c@{\,\,\,\,}c@{\,\,\,\,}c@{\,\,\,\,}:c@{\,\,\,\,}c@{\,\,\,\,}:c@{\,\,\,\,}c@{}}0&0&0&0&I&0&0&0\\\
0&0&0&0&0&I&0&0\end{array}\right],~{}U=\small\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}:c@{\,\,\,\,\,\,}c@{\,\,\,\,\,\,}:c@{\,\,\,\,\,\,}c@{}}I&0&0&0&0&0&0&0\\\
0&I&0&0&0&0&0&0\\\ 0&0&I&0&0&0&0&0\\\ 0&0&0&I&0&0&0&0\\\ \hdashline
0&0&0&0&0&0&I&0\\\ 0&0&0&0&0&0&0&I\end{array}\right],$

can be written as condition (i) of the FBSP in Theorem 5. This shows that (31)
holds if and only if both (29) and (30) in Theorem 2 hold, as a result of the
FBSP in Theorem 5333The advantage of introducing an interconnection channel
lies in the reduction of the controller synthesis computational effort that is
obtained by imposing a particular structure on the multipliers $Q,R,S$ and
decomposing the synthesis equations at the price of introducing some
conservatism..  $\blacksquare$

###  IV-B Extended FBSP in Dual Form

###### Theorem 3

Extended FBSP in Dual Form The well-posed rational interconnected system in
(7) has the $\mathcal{L}_{2}$-gain from $w$ to $z$ smaller than $\gamma$ if
and only if the following statement holds. There exists
${Y}=\tilde{F}Y^{-1}\tilde{F}^{\top}={Y}^{\top}>0$, with $\tilde{F}$ non-
singular and multiplier matrices
$\tilde{Q}=\tilde{Q}^{\top},\tilde{R}=\tilde{R}^{\top},\tilde{S}$, such that
(34) and (35) hold.

$\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{\top}\left[\begin{array}[]{@{}c@{\,\,\,\,\,\,}c@{}}\tilde{Q}&\tilde{S}\\\
\tilde{S}^{\top}&\tilde{R}\end{array}\right]\left[\begin{array}[]{@{}c@{}}I\\\
\mathcal{P}\end{array}\right]<0,$ (34)

$\small\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{\top}\small\left[\begin{array}[]{@{}c@{}
@{\,\,\,}c@{\,\,\,}c@{\,\,\,}c@{\,\,\,} : @{\,\,\,}c@{\,\,\,\,} c@{\,\,\,\,} :
@{\,\,\,\,}c@{\,\,\,}c@{\,\,\,}}0&0&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}}&0&0&0&0&0\\\
0&0&0&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}}&0&0&0&0\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}^{\top}}&0&0&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}}-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{Y}}&0&0&0&0\\\
0&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}^{\top}}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}^{\top}}-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{Y}}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{F}^{\top}}&0&0&0&0\\\
\hdashline 0&0&0&0&\tilde{Q}&\tilde{S}&0&0\\\
0&0&0&0&\tilde{S}^{\top}&\tilde{R}&0&0\\\ \hdashline
0&0&0&0&0&0&-\frac{1}{\gamma^{2}}I&0\\\
0&0&0&0&0&0&0&I\end{array}\right]\small\left[\begin{array}[]{@{}c@{\,\,}@{\,\,}c@{}@{\,\,}:c@{}:c@{}
}-\mathcal{A}^{\top}&0&-\mathcal{C}_{1}^{\top}&-\mathcal{C}_{2}^{\top}\\\
-\mathcal{A}^{\top}&0&-\mathcal{C}_{1}^{\top}&-\mathcal{C}_{2}^{\top}\\\
I&0&0&0\\\ 0&I&0&0\\\
\hdashline-\mathcal{B}_{1}^{\top}&0&-\mathcal{D}_{11}^{\top}&-\mathcal{D}_{12}^{\top}\\\
0&0&I&0\\\
\hdashline-\mathcal{B}_{2}^{\top}&0&-\mathcal{D}_{21}^{\top}&-\mathcal{D}_{22}^{\top}\\\
0&0&0&I\end{array}\right]>0$ (35)

  * Proof:

We start from the primal formulation of the EFBSP in Theorem 2 and we exploit
the Dualization Lemma reported in Lemma 4.8 from [17] in which the middle
matrices in (30) and in (35) play the same roles as $P$ and $P^{-1}$ in Lemma
4.8 from [17], respectively. Then, by considering the dual system with respect
to the system in (32), i.e.

$\begin{cases}\dot{x}(t)&=\mathcal{A}^{\top}(\mathcal{P})x(t)+\mathcal{C}_{1}^{\top}(\mathcal{P})w(t)\\\
z(t)&=\mathcal{B}_{1}^{\top}(\mathcal{P})x(t)+\mathcal{D}_{11}^{\top}(\mathcal{P})w(t),\end{cases}$
(36)

the dual formulation as in (34)-(35) is obtained and this concludes the proof.
$\blacksquare$

##  V Decomposed Extended FBSP

In this section we present the main result of this paper, which is a
decomposed EFBSP for scalable interconnected controller synthesis, given in
the following theorem.

###### Theorem 4

Decomposed EFBSP in Dual Form: The well-posed rational interconnected system
in (32), whose interconnection matrix $\mathcal{P}={P}\otimes{I_{N}}$ is
symmetric, has $\mathcal{L}_{2}$-gain from $w$ to $z$ smaller than $\gamma$ if
there exist $Y^{d}$, $Y^{i}$, $\tilde{F}^{d}$ non-singular, such that
${Y}=\tilde{F}Y^{-1}\tilde{F}^{\top}={Y}^{\top}>0$, with
$Y={I}\otimes{Y^{d}}+{P}\otimes{Y^{i}}$,
$\tilde{F}={I_{N}}\otimes{\tilde{F}^{d}}$ and there exist matrices
$\tilde{Q}^{i}$, $\tilde{Q}^{d}$, $\tilde{R}^{i}$, $\tilde{R}^{d}$,
$\tilde{S}^{d}$, $\tilde{S}^{i}$, with $\tilde{Q}^{d}=\tilde{Q}^{d\top}$,
$\tilde{R}^{d}=\tilde{R}^{d\top}$, and $M^{d}$, such that (37) and (38) hold,
where $\lambda_{h}\in\mathrm{spec}({{P}})$, $h=1,\dots,N$, are the eigenvalues
of $P$.

$\tilde{Q}^{d}+\lambda_{h}(\tilde{Q}^{i}-\tilde{S}^{d\top}-\tilde{S}^{d})+\lambda_{h}^{2}(-\tilde{S}^{i\top}-\tilde{S}^{i}+\tilde{R}^{d})+\lambda_{h}^{3}\tilde{R}^{i}<0\quad\forall\lambda_{h}\in\mathrm{spec}({P}),$
(37)

$\displaystyle\small\left[\begin{array}[]{@{}c@{}}\star\end{array}\right]^{\top}\scriptsize\left[\begin{array}[]{@{}c@{\\!\\!\\!}
@{\,\,\,\,\,\,\,\,\,\,}c@{\,\,\,\,}c@{\,\,\,\,}c@{\\!\\!\\!} :
@{\,\,\,\,\,}c@{\,\,\,} c@{\,\,\,} : @{\,\,}c@{\,\,\,}c@{}}0&0&I&0&0&0&0&0\\\
0&0&0&I&0&0&0&0\\\ I&0&0&\tilde{F}^{d}-{Y}^{d}-\lambda_{h}{Y}^{i}&0&0&0&0\\\
0&I&\tilde{F}^{d\top}-{Y}^{d}-\lambda_{h}{Y}^{i}&\tilde{F}^{d}+\tilde{F}^{d\top}&0&0&0&0\\\
\hdashline
0&0&0&0&\tilde{Q}^{d}+\lambda_{h}\tilde{Q}^{i}&\tilde{S}^{d}+\lambda_{h}\tilde{S}^{i}&0&0\\\
0&0&0&0&\tilde{S}^{d\top}+\lambda_{h}\tilde{S}^{i\top}&\tilde{R}^{d}+\lambda_{h}\tilde{R}^{i}&0&0\\\
\hdashline 0&0&0&0&0&0&-\frac{1}{\gamma^{2}}I&0\\\
0&0&0&0&0&0&0&I\end{array}\right]\footnotesize\left[\begin{array}[]{@{\,}c@{\,\,\,\,\,}c@{\,\,\,\,\,}:c@{\,\,\,\,\,}:c@{\,\,\,\,\,}
}-(A_{h}\tilde{F}^{d}+B_{u_{h}}M^{d})^{\top}&0&-({C}_{q_{h}}\tilde{F}^{d}+D_{qu_{h}}M^{d})^{\top}&-({C}_{z_{h}}\tilde{F}^{d}+D_{zu_{h}}M^{d})^{\top}\\\
-(A_{i}\tilde{F}^{d}+B_{u_{h}}M^{d})^{\top}&0&-({C}_{q_{h}}\tilde{F}^{d}+D_{qu_{h}}M^{d})^{\top}&-({C}_{z_{h}}\tilde{F}^{d}+D_{zu_{h}}M^{d})^{\top}\\\
I&0&0&0\\\ 0&I&0&0\\\
\hdashline-\mathcal{B}_{1_{h}}^{\top}&0&-\mathcal{D}_{{11}_{h}}^{\top}&-\mathcal{D}_{{12}_{h}}^{\top}\\\
0&0&I&0\\\
\hdashline-\mathcal{B}_{2_{h}}^{\top}&0&-\mathcal{D}_{{21}_{h}}^{\top}&-\mathcal{D}_{{22}_{h}}^{\top}\\\
0&0&0&I\end{array}\right]$ $\displaystyle>0,$ (38)
$\displaystyle\forall{h=1,\dots,N}$

  * Proof:

Consider (34) and (35) in Theorem 3, where the closed-loop matrices of the
interconnected system under the distributed controller are as in (III). We
perform the following variable substitution in order to convexify (35)

$M=\hat{K}\tilde{F}=\begin{bmatrix}K^{d}\\\ K^{i}\end{bmatrix}\tilde{F}$ (39)

leading to the closed-loop matrices in (III). We note that these closed-loop
matrices all have a block-diagonal structure. Next, choosing the multiplier
matrices in (34) and (35) structured as

$\displaystyle\tilde{Q}={I}\otimes{\tilde{Q}^{d}}+{P}\otimes{\tilde{Q}^{i}},$
$\displaystyle\quad\tilde{S}={I}\otimes{\tilde{S}^{d}}+{P}\otimes{\tilde{S}^{i}},$
(40) $\displaystyle\tilde{R}={I}\otimes{\tilde{R}^{d}}$
$\displaystyle+{P}\otimes{\tilde{R}^{i}},$

we apply the following algebraic transformation. Let $Z$ be an orthonormal
matrix such that $Z^{\top}PZ=\Lambda$, where $\Lambda$ is the diagonal matrix
having the eigenvalues of $P$ as diagonal entries, i,e.
$\Lambda=diag_{i=1}^{N}\\{\lambda_{h}\\}$. Then, we apply the algebraic
transformations

$\displaystyle\hat{E}_{MC}$ $\displaystyle=\hat{Z}^{\top}\,E_{MC}\,\hat{Z}$
$\displaystyle\hat{E}_{NC,1}^{\top}\hat{E}_{NC,2}\hat{E}_{NC,1}$
$\displaystyle=\hat{H}^{\top}\,E_{NC,1}^{\top}\,\hat{T}\,\hat{T}^{\top}\,E_{NC,2}\,\hat{T}\,\hat{T}^{\top}\,E_{NC,1}\,\hat{H},$
(41)

where $E_{MC}$ is the multiplier condition in (34), and $E_{NC,1}$ and
$E_{NC,2}$ are the matrices that form the nominal condition
$E_{NC,1}^{\top}\,E_{NC,2}\,E_{NC,1}$ in (35), respectively. The
transformation in (Proof:) is defined by the matrices $\hat{Z}:=Z\otimes
I_{z_{1}}$, $\hat{H}=diag_{i=1}^{3}\\{\hat{H}_{i}\\}$, with
$\hat{H}_{i}:=Z\otimes I_{h_{i}}$, and
$\hat{T}=diag_{i=1}^{3}\\{\hat{T}_{i}\\}$ with $\hat{T}_{h}:=Z\otimes
I_{t_{i}}$, $i=1,2,3$, with $z_{1},h_{1},h_{2},h_{3},t_{1},t_{2},t_{3}$ being
the dimensions of the identity matrices such that the block-partitions of the
transformation matrices $\hat{Z}$, $\hat{H}$ and $\hat{T}$ matches the block-
partitions of the matrices $E_{MC}$, $E_{NC,1}$ and $E_{NC,2}$ as indicated in
(34) and (35), respectively, to which they are applied.

After some basic algebraic manipulations and exploiting the properties of the
Kronecker product, it can be seen that the block-diagonal blocks are only
multiplied with identity matrices, and that the matrix $P$ is diagonalized,
resulting in LMIs (37) and (38).  $\blacksquare$

In comparison with the decomposed controller synthesis approach in [6], where
the block-diagonal structure needs to be imposed on the multiplier matrices
$\tilde{Q},\tilde{R},\tilde{S}$ and on the Lyapunov matrix $Y$ in order to
decompose the synthesis conditions in the FBSP, imposing the structure in (40)
in the EFBSP allows us to obtain the same degree of decomposition and thus
scalability, i.e., to decompose the synthesis conditions into small matrix
inequalities in the size of the individual subsystems, with the advantage of
introducing less conservatism. This is achieved by exploiting the
diagonalizability of the matrix $P$. Moreover, since $P$ is symmetric, the
matrix $Z$ that diagonalizes $P$ is orthonormal and consequently the value of
$\gamma$ obtained from (37)-(38) does not differ from the one for the original
system. More details related to the case in which $P$ is not symmetric are
discussed in [15].

###### Remark 2

The computational time needed to solve the LMIs (37)-(38) scales linearly,
while the one needed to solve (34)-(35) scales polynomially with $N$.
Moreover,  at the price of introducing some conservatism, by imposing the
additional constraints $R^{i}=0$ and
$-\tilde{S}^{i\top}-\tilde{S}^{i}+\tilde{R}^{d}>0$ in Theorem 4, the
constraints in (37) become quadratic and convex in the variables
$\lambda_{h}$. Therefore one can simply solve it only for the constraints
corresponding to the largest and smallest eigenvalues of $P$.

###### Remark 3

The structure imposed on the optimization variables
$Y,\tilde{Q},\tilde{R},\tilde{S}$ involved in the EFBSP can also be chosen as
$Y={I_{N}}\otimes{Y^{d}}+{P}\otimes{Y^{i}}+{P_{1}}\otimes{Y^{i}_{1}}+\dots+{P_{r}}\otimes{Y^{i}_{r}}$
and the same for $\tilde{Q},\tilde{R},\tilde{S}$, in (35) and (34), where
$P,P_{1},\dots P_{r}$ are simultaneously diagonalizable pattern matrices, see
[18] and [15] for more details about simultaneously diagonalizable matrices
and their applications. This approach can further reduce conservatism since
more non-zero entries, i.e., more degrees of freedom for
$Y,\tilde{Q},\tilde{R},\tilde{S}$, are allowed. Let $P_{1}$ be the matrix with
scalar entries $p_{ij}$, i.e., $P_{1}=[p_{ij}]_{i,j=1,\dots,N}$. Let us
suppose that $P_{1}$ is the pattern matrix, simultaneously diagonalizable with
$P$, with the highest number of non-zero entries, and hence the matrix
allowing the less restrictive structure on $Y,\tilde{Q},\tilde{R},\tilde{S}$
among all the simultaneously diagonalizable matrices with $P$,
$P_{1},P_{2},\dots,P_{r}$. $P_{1}$ may be found as the solution of the
following optimization problem:

$P_{1}=\max_{\begin{subarray}{c}s.t.\\\ P_{1}=P_{1}^{T}\\\ PP_{1}=P_{1}P\\\
p_{ii}=0\\\ p_{ij}(1-p_{ij})=0\end{subarray}}\sum_{i<j}p_{ij}.$ (42)

where the constraint $PP_{1}=P_{1}P$ fully characterizes the simultaneous
diagonalizability [18].

##  VI Example and Simulation Results

By referring to the description in (4)-(5), we consider a system whose
subsystems are described by the following matrices:

$\displaystyle A^{d}$ $\displaystyle=\begin{bmatrix}0&-0.2&0.8\\\
-0.9&-0.7&-0.4\\\ -0.9&0.5&-0.6\end{bmatrix},{B}^{d}_{u}=\begin{bmatrix}0.2\\\
0.2\\\ -0.1\end{bmatrix}{B}^{d}_{{w}}=\begin{bmatrix}0.5\\\ 0.5\\\
0.5\end{bmatrix}$ $\displaystyle{A}^{i}$
$\displaystyle=\begin{bmatrix}0.2&0.1&-0.1\\\ -0.4&-0.1&-1.0\\\
-0.3&-0.1&0.0\end{bmatrix},{C}^{d}_{y}=\begin{bmatrix}1.1&-2.1&-1.6\\\
-0.3&-0.4&0.5\\\ 0.7&-0.8&0.3\end{bmatrix},$ $\displaystyle{C}^{d}_{{{z}}}$
$\displaystyle=\begin{bmatrix}0.8&0.3&0.3\\\
-0.1&0.1&-0.1\end{bmatrix},{D}^{d}_{{{z}u}}=\begin{bmatrix}0.1\\\
0.9\end{bmatrix},{C}^{i}_{{{z}}}=\begin{bmatrix}0&0&0\\\
0.3&0.2&0.5\end{bmatrix},$

all the others are set to zero. The subsystems are interconnected according to
the pattern matrix

$P=\begin{bmatrix}0&1&0&0&1&0\\\ 1&0&1&1&0&1\\\ 0&1&0&0&1&0\\\ 0&1&0&0&1&0\\\
1&0&1&1&0&1\\\ 0&1&0&0&1&0\end{bmatrix}.$

Moreover, the interconnection variables are defined as
$p_{h}=[x_{h}^{i\top},u_{h}^{i\top}]^{\top}$ where $x_{h}^{i}$ and $u_{h}^{i}$
are the interconnected state from and control action depending on neighboring
subsystem $G_{j}$ of subsystem $G_{h}$.

Results. In the following some numerical results are shown in order to
illustrate the proposed controller synthesis via decomposed EFBSP in Theorem
4. Table I shows a comparison between the results obtained by applying the
FBSP in which all the optimization variables are restricted to be block-
diagonal and the results obtained by applying the EFBSP in which the matrix
$\tilde{F}$ is restricted to be block-diagonal, while the Lyapunov matrix $Y$
and the multipliers are structured as in (40). The optimization problem has
been coded in $\mathtt{Matlab}$ and solved through the non-linear solver
$\mathtt{Penbmi}$ from the $\mathtt{Tomlab}$ toolbox. Since in our simulations
the solver did not converge, an additive constraint that restricts $\gamma$ to
be lower than the value of $\gamma$ obtained in the standard FBSP with all the
optimization variables restricted to have block-diagonal structure has been
added.

| FBSP | EFBSP
---|---|---
|  Y blkdiag $\tilde{Q},\tilde{R},\tilde{S}$ blkdiag |  $\tilde{F}$ blkdiag, $(Y,\tilde{Q},\tilde{R},\tilde{S})_{\otimes}$
$\gamma$ | 10.2533 | 9.2236
$\mathcal{H_{\infty}}$ | 10.1814 | 8.0553

TABLE I: Numerical results

It is worth saying that, unlike the EFBSP, relaxing the block-diagonal
structure of the multiplier matrices in the standard FBSP to the structure in
(40) does not bring any improvement in terms of performance and this is one of
the reasons why by means of the EFBSP better performance may be reached.

## References

  * [1] N. Matni and A. Rantzer  “Low-rank and low-order decompositions for local system identification”  In _ArXiv_ , 2014  DOI: 1403.7175
  * [2] M. Mesbahi and F.Y. Hadaeg  “Formation fluing control of multiple spacecraft via graphs, matrix inequalities, and switching”  In _AIAA Journal of Guidance, Control, and Dynamics._ 22.24, 2001, pp. 369–377  DOI: 10.1109/CCA.1999.801145
  * [3] M.R. Jovanović and B. Bamieh  “On the ill-posedness of certain vehicular platoon control problems”  In _IEEE Transactions on Automatic Control_ 9.50, 2005, pp. 1307–1321  DOI: 0.1109/TAC.2005.854584
  * [4] Yvonne Rebecca Stürz, Annika Eichler and Roy Stephen Smith  “A framework for distributed control based on overlapping estimation for cooperative tasks”  In _IFAC-PapersOnLine_ 50.1 Elsevier B.V., 2017, pp. 14296–14301  DOI: 10.1016/j.ifacol.2017.08.1863
  * [5] J. D. Wolfe, D. F. Chichka and J. L. Speyer  “Decentralized controllers for unmmaned aerial vehicle formation flight”  In _Amer. Inst. Aeronautics Astronautics_ , 1996
  * [6] P. Massioni and M. Verhaegen  “A full block S-Procedure application to distributed control”  In _American Control Conference_ , 2010, pp. 2338–2343  DOI: 10.1109/ACC.2010.5530556
  * [7] H. Raza and P. Ioannou  “Vehicle following control design for automated highway systems”  In _1997 IEEE 47th Vehicular Technology Conference. Technology in Motion_ 2, 1997, pp. 904–908  DOI: 10.1109/VETEC.1997.600460
  * [8] L.E. Parker  “Current state of the art in distributed autonomous mobile robotics”  In _Distributed Autonomous Robotic Systems 4 (DARS)_ , 2000
  * [9] Yvonne Rebecca Stürz, Annika Eichler and Roy Stephen Smith  “Fixed mode elimination by minimum communication within an estimator-based framework for distributed control”  In _IEEE Control Syst. Lett._ 1.2, 2017, pp. 346–351
  * [10] Cédric Langbort, Ramu Sharat Chandra and Raffaello D’Andrea  “Distributed control design for systems interconnected over an arbitrary graph”  In _IEEE Trans. Automat. Contr._ 49.9, 2004, pp. 1502–1519  DOI: 10.1109/TAC.2004.834123
  * [11] P. Massioni  “Distributed control for alpha-heterogeneous dynamically coupled systems”  In _Systems & Control Letters_ 72, 2014, pp. 30–35  DOI: 10.1109/CDC.2013.6760306
  * [12] Yvonne Rebecca Stürz, Annika Eichler and Roy Stephen Smith  “Distributed control design for heterogeneous interconnected systems”  In _ArXiv:2004.04876_ , 2020  DOI: 2004.04876
  * [13] Y.R. Stürz, A. Eichler and R. Smith  “Scalable controller synthesis for heterogeneous interconnected systems applicable to an overlapping control framework”  In _European Control Conference_ , 2018, pp. 2561–2568  DOI: 10.23919/ECC.2018.8550328
  * [14] M. Farhoodi and T. H. Beheshti  “Extended $H_{2}$, $H_{\infty}$ and pole placement characterization for continuous-time systems”  In _Annual IEEE India Conference_ , 2008, pp. 536–541  DOI: 10.1109/INDCON.2008.4768781
  * [15] P. Massioni and M. Verhaegen  “Distributed control for identical dynamically coupled systems: a decomposition approach”  In _IEEE Transactions on Automatic Control_ 1.54, 2009, pp. 124–135  DOI: 10.1109/TAC.2008.2009574
  * [16] C. W. Scherer  “LPV control and full block multipliers”  In _Automatica_ 37.3, 2001, pp. 361–375
  * [17] C. Scherer and S. Weiland  “Linear Matrix Inequalities in Control”  Retrieved from http://een.iust.ac.ir, 2005
  * [18] R.A. Horn and C.R. Johnson  “Matrix Analysis”  Cambridge University Press, Cambridge, 1985

## APPENDIX

###  VI-A Full Block S-Procedure

###### Theorem 5

[16] Assume $U\in\mathbb{R}^{p\times n},V\in\mathbb{R}^{q\times
n},W\in\mathbb{R}^{n\times n}$, with $W=W^{\top}$, and let $\mathcal{I}$ be a
subspace of $\mathbb{R}^{n}$. Furthermore, let
$\mathcal{S}(\rho)\in\mathbb{R}^{q}$ be a subspace that depends continuously
on the parameter $\rho$ which varies in the compact path-connected set
$\bm{\rho}$. Let $\mathcal{B}(\rho)$ be a family of subspaces of
$\mathbb{R}^{n}$, defined as

$\mathcal{B}(\rho):=\\{x\in\mathcal{I}|Vx\in\mathcal{S}(\rho)\\}$

Then the following two statements are equivalent:
(i) $U^{\top}WU<0$ on $\mathcal{B}(\rho)$, $\forall\rho\in\bm{\rho}$.
(ii) There exists $P_{e}=P_{e}^{\top}>0$ on $\mathcal{S}(\rho)$,
$\forall\rho\in\bm{\rho}$, which satisfies $U^{\top}WU+V^{\top}P_{e}V<0$ on
$\mathcal{I}$.