論文 / 著書情報
Article / Book Information
題目(和文)

動的ネットワークシステムにおける制御器群の分散設計

Title(English)

Distributed Design of Controllers in Dynamical Network Systems

著者(和文)

笹原帆平

Author(English)

Hampei Sasahara

出典(和文)

学位:博士(工学),学位授与機関:東京工業大学,
報告番号:甲第11128号,
授与年月日:2019年3月26日,
学位の種別:課程博士,
審査員:井村 順一,藤田 政之,三平 満司,山北 昌毅,早川 朋久,畑中 健志

Citation(English)

Degree:Doctor (Engineering),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第11128号,
Conferred date:2019/3/26,
Degree Type:Course doctor,
Examiner:,,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Tokyo Institute Research Repository)

Distributed Design of Controllers
in Dynamical Network Systems

Hampei Sasahara
School of Engineering
Tokyo Institute of Technology

This dissertation is submitted for the degree ofDoctor of Engineering

February 2019Abstract
This thesis provides a line of work on distributed design of structured controllers forlarge-scale dynamical network systems. Although most existing distributed controller
design methods are built on the premise that the controllers are designed and integrated
in a centralized manner, actual network systems are managed by multiple operators inmany cases and each subcontroller is independently designed by its own operator based
only on the model of the corresponding subsystem. In this thesis, novel distributed
design methods for decentralized controllers and distributed controllers are proposed.
An approach based on retrofit control is proposed for distributed design of decentralized
controllers. The key idea is to regard the network system as an interconnected systemamong a subsystem of interest and the others and to attach a controller that preserves
stability of the original network system. It is revealed that all retrofit controllers havea structure that rectifies the effect from the interconnection signal to the measurement
signal. Moreover, to confirm the effectiveness of the proposed method, a numericalexample through a benchmark model representing the bulk power system in the eastern
half of Japan is shown. In addition, the results of retrofit control are generalized to
glocal control where global and local controllers are designed in a distributed manner.
Based on hierarchical representation, which is an equivalent representation of theoriginal network system, a novel glocal control method is proposed.

Table of contents
1 Introduction
1.1 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Contributions of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
4

2 Distributed Design of Local Controllers via Retrofit Control
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Retrofit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Definition of Retrofit Controllers . . . . . . . . . . . . . . . . .
2.3 Characterization of All Retrofit Controllers . . . . . . . . . . . . . . . .
2.3.1 Characterization of All Retrofit Controllers . . . . . . . . . . . .
2.3.2 Closed-Loop Map with Retrofit Controller and Difficulty on
Retrofit Controller Design . . . . . . . . . . . . . . . . . . . . .
2.4 Design of Output-Rectifying Retrofit Controllers with Interconnection
Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Definition of Output-Rectifying Retrofit Controllers and InputRectifying Retrofit Controllers . . . . . . . . . . . . . . . . . . .
2.4.2 Parameterization of All Output-Rectifying Retrofit Controllers
with Interconnection Signal . . . . . . . . . . . . . . . . . . . .
2.5 Dual Results in Input-Rectifying Retrofit Controllers . . . . . . . . . .
2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9
10
10
11
12
13
16
18
18
19
23
26

3 Output-Rectifying Retrofit Controllers without Interconnection Signal
27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vi
3.3
3.4
3.5

Parameterization of All State-Feedback Output-Rectifying Retrofit Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dual Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28
35
37

4 Distributed Design of Power System Stabilizers in Power Grids withRenewable Energy Resources
39
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Modeling of PV-Integrated Power Systems . . . . . . . . . . . . . . . . 40
4.2.1 Synchronous Generator . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.2 Load, PV Power Plant, and Non-Unit Bus . . . . . . . . . . . . 46
4.2.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 PSS Design based on Retrofit Control . . . . . . . . . . . . . . . . . . . 48
4.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.2 Hierarchical State-Space Expansion for Retrofit Control in TimeDomain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.3 Proposed PSS Design Method . . . . . . . . . . . . . . . . . . . 52
4.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Glocal Control of Network Systems via Hierarchical Representation 65
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Hierarchical Representation . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 Definition of Hierarchical Representation . . . . . . . . . . . . . 68
5.3.2 Characterization of Hierarchical Representation . . . . . . . . . 69
5.4 Glocal Control for Network Systems . . . . . . . . . . . . . . . . . . . . 74
5.4.1 Difficulty on Glocal Controllers Implementation . . . . . . . . . 74
5.4.2 Rectifier Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5 Application to Power System Control . . . . . . . . . . . . . . . . . . . 78
5.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 Conclusion
81
6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Bibliography

83

Chapter 1
Introduction
1.1

Backgrounds

The thesis discusses control of large-scale dynamical network systems. A dynamical
network system is composed of multiple dynamical subsystems and a network connecting
the subsystems. Behavior of each subsystem’s states obeys not only its own dynamics
but also interaction among interconnected subsystems through the network. The notionof large-scale dynamical network systems covers a great variety of systems [1] andtypical examples include power systems [2, 3], transportation systems [4], and social
networks [5–7]. Safely operating such a large-scale network system with reliable control
is an important issue because when a contingency happens at a part of the network
system the failure widely spreads over the whole. In fact, serious blackouts causedby the cascading failure often happen throughout the world, such as the blackout of
North America in 2003 where the number of affected people reaches into 45 million [8].
Although a great deal of effort has been devoted to the research of controller
design in large-scale dynamical network systems, the problem still remains to bechallenging. Several difficulties are arisen when controlling large-scale network systems
and researchers in the filed of control engineering have mainly focused on the problem
of information structure inside the designed controllers [9]. In the typical setting, it
is supposed that a subcontroller is assigned to every subsystem and these controllers
manage the whole network system in a coordinated manner by exchanging signal
information with respect to its corresponding subsystem through communication with
one another. Owing to the information exchange way, the structure of the controllers is
restricted and hence the feasible set of designable controller is also limited. As a result,the mathematical problem of controller design is different from the classic one where
no restriction on controller structure is placed. Several controller design methods, e.g.,2decentralized control and distributed control [10, 11], have been proposed to deal withinformation structure and the effectiveness has been verified.
Based on the existing results, the next issue to be resolved is distributed designof controllers [12], where each controller is designed in a distributed manner. In fact,
most existing distributed controller design methods are built on the premise that thecontrollers are designed and integrated in a centralized manner; in other words, it is
implicitly supposed that a single system operator manages the entire network system anddesigns all subcontrollers in the network system. Actual network systems, however, aremanaged by multiple operators in many cases and each subcontroller is independently
designed by its own operator based only on the model of the corresponding subsystem.
For instance, in power systems, multiple independent system operators exist and eachof them is responsible for achieving supply-demand balance in its region while keeping
the stability of the entire grid by subcontrollers, such as power system stabilizers(PSSs), which are designed independently [13]. Consequently, it is necessary to develop
a distributed design method of controllers to deal with difficulties in real network
systems.
The notion of distributed design has been proposed in [12] and the literature on thisissue is still in its infancy. A distributed design method has been proposed in [12], but it
places the unrealistic assumption that independent control inputs can be injected into
every differential equation in the system. Although in [14] the authors have extended theresult to the case of fully-actuated systems, the applicability is still limited. As anotherconcept achieving distributed design, controller design methods based on system level
synthesis [15–17] have been proposed. The key idea is to confine the closed-loop map
to its local region and to make the optimal controller design problem separable andsolvable locally. The approach can be applied only to discrete-time systems essentiallyand it cannot be determined in advance whether the optimization problem in thedesign process is feasible or not. In [18], a heuristic approach for designing intelligent
balancing authorities that control subsystems inside each corresponding area to depress
dynamic response to contingencies of interest is proposed. Although its effectiveness isshown numerically, no theoretical results are derived.
As a novel distributed design method of decentralized controllers, retrofit control [19,20] has been recently proposed. The basic operation of retrofit control is to regard
the system as an interconnected system among a subsystem of interest and the others
and to design a retrofit controller that guarantees the entire system stability for any
possible neighboring subsystems. Every operator in the targeted network system can
independently design and implement a decentralized subcontroller while preserving

3
the stability of the whole network system by employing retrofit controllers, which
are designed based only on the model of the corresponding subsystem. Retrofit
control requires no communication among sub-controllers, which has a potential of
applicability to a large-class of interconnected systems, because all subcontrollers are
purely decentralized and no lines are needed among the subcontrollers.

1.2

Contributions of Thesis

In view of the backgrounds mentioned in Section 1.1, the following problems are
considered in this thesis.
First, we consider distributed design problem of decentralized controllers for alarge-scale network system that is stably operated. There are multiple operatorswith the aim of suppressing effects of disturbance injected into the network systemby implementing subcontrollers. Each operator can attach a subcontroller to thecorresponding subsystem but the subcontroller design is performed based only on thesubsystem model without the other subsystems and the network, called environment.
The difficulty here is that existing controller design methods cannot deal with the
unknown environment and even stability of the resulting system cannot be guaranteedwith traditional ones. Under the situation, a controller design method for decentralized
controllers in a distributed fashion that can assure the stability of the whole network
system is proposed via retrofit control in this thesis.
Second, we consider the PSS design problem for frequency control in power gridswith renewable energy resources. As a practical problem, stabilization of power grids towhich large amount of distributed energy resources is introduced has been increasingly
important. The frequencie of every synchronous generator in a large power grid isnecessarily kept to be around the nominal value; otherwise these lose synchronism
and a fatal fault, e.g., blackouts, occurs. Thus, it is extremely important to carefully
design PSS that has the role to maintain the frequencies around the nominal value
by measuring the corresponding generator’s frequency and voltage and controllingthe voltage injected into the automatic voltage regulator (AVR). Although existing
design methods, e.g., a single-machine infinite-bus model [21], where the dynamics
of the other parts is perfectly ignored for the controller design, work well for the
traditional power grids, for future power grids where a large amount of renewableenergy resources is installed controller design methods ought to be established via
distributed design which gives assurance of the internal stability of the whole grids
in a theoretical manner [22–24]. A numerical simulation is shown to demonstrate the4effectiveness of our proposed approach for PSS design in power grids with renewableenergy resources through IEEJ EAST 30-machine power system (EAST30) [25] withphotovoltaic (PV) plants.
Finally, we consider the problem of integration of supervising controller designand decentralized local controller design for network systems. We refer to such aframework where a global controller spatially broadcasts the control inputs to theentire system and local decentralized controllers are simultaneously implemented as
glocal control. The concept of glocal control has been already introduced in [26] but
a systematic design of glocal controllers has not been established yet. Generalizing
the results obtained for the above problems, we propose a novel glocal control that
achieves distributed design of the controllers.

1.3

Organization

This thesis is organized as follows: First, in Chapter 2, a distributed design strategy ofdecentralized controllers via a newly defined retrofit control approach is proposed. To
treat the network system mentioned in the previous section, we model the network
system as an interconnected system with a valid model and an unknown environmentthat is assumed to be a system with which the entire system is stable. Under thesetting, retrofit controllers are defined as the controllers that stabilize the interconnected
system for any possible environment. A necessary and sufficient condition for retrofit
controllers is derived based on Youla parameterization in an implicit form. Next,
we define two kinds of retrofit controllers, output-rectifying retrofit controllers andinput-rectifying retrofit controllers, to derive an explicit paramaterization of retrofit
controllers. An explicit parameterization of all output-rectifying retrofit controllers
is given under the assumption that the interconnection signal from the environmentis available. It is found out that the structure inside all output-rectifying retrofit
controllers is the proposed one in [19]. Owing to the parameterization, an optimal
retrofit controller can be designed and consequently distributed design of decentralized
controllers is achieved. Moreover, we show that similar structure is observed inside
input-rectifying retrofit controllers as dual results.
In practical network systems, it is desirable to design a retrofit controller without themeasurablity assumption on the interconnection signal because implementing sensors to
obtain the interconnection signal often costs and it is sometimes impossible to measure
the interconnection signal. In Chapter 3, we aim at relaxing the assumption andgeneralizing the result in Chapter 2. We consider the case where interconnection signal

5
is unavailable and derive a parameterization of all state-feedback retrofit controllers
without interconnection signal. It is shown that a similar structure can be observed inall state-feedback retrofit controllers to that of retrofit controllers with interconnection
signals.
In Chapter 4, for the purpose of treating the stabilization problem for power grids,
we first give a mathematical model of EAST30 composed of synchronous machines,
loads, PV generators in addition to buses that form the network. The dynamics withrespect to synchronous machines is a 13-dimensional dynamical system each of whichis composed of a synchronous generator, an excitation system with AVR, and a turbinewith governor. The loads and PV power plants are modeled as constant loads andconstant power sources, respectively. We show that the power grid can be regarded as
an interconnected system treated in Chapter 2 and 3 and apply the proposed approachfor PSS design. A numerical simulation is shown to demonstrate the effectiveness of
our proposed approach for PSS design in power grids with renewable energy resources
through the PV-integrated EAST30 model [25].
We generalize the above approach to glocal (global/local) control, where spatiallydistributed decentralized controllers and a central broadcasting controller cooperatively
manage the entire network system, in Chapter 5. A novel glocal control method basedon hierarchical representation, whose notion is introduced in this work, is proposed.
Given the definition of hierarchical representation, we derive a necessary and sufficientcondition for existence of hierarchical representation for a network system. Based on
the hierarchical representation, we propose a linear function observer that preserves thehierarchical structure and distributed design of global and local controllers is achieved
with the observer. The design procedure of controllers are independently performed
and it indicates that the controllers can be designed in a distributed manner. It isshown that both of local and global behaviors are simultaneously suppressed thanks tothe existence of the global controller. A numerical example with EAST30 is shown toillustrate the effectiveness of the proposed glocal control method.
Finally, Chapter 6 draws conclusion.

6
NOTATION: The following notation is to be used through the thesis:R
the set of real numbersCthe set of complex numbers
L2
the set of all square integrable functions
n×m
R
the set of real and rational n × m transfer matrices
RP n×mthe set of proper transfer matrices in Rn×m
RHn×m
the set of stable transfer matrices in RP n×m∞|I|
the cardinality of a set I
dim(X )
the dimension of a vector space X
X⊥
the orthogonal subspace of a subspace X
X +Y
the sum space of subspaces X and Yjthe imaginary unit∗x
the conjugate of x ∈ C|x|the absolute value of x ∈ C∠xthe angle of x ∈ C
Re x (Im x)
the real (imaginary) value of x ∈ C
comp(x)[1 j]x for x ∈ R2∥x∥
the Euclidean norm of a vector xcol(xi )i∈I
the vector where xi for i ∈ I are arranged in verticalInthe n-dimensional identity matrix
TM
the transpose of a matrix MM†a pseudo inverse of a full rank matrix M
im M
the image of a matrix M
ker M
the kernel of a matrix M∥M ∥the maximum singular value of a matrix M
M ⊗N
the Kronecker product of matrices M and N
σ (M )
the spectrum of a matrix M
det(M )
the determinant of a matrix M
diag(M1 , . . . , Mn ) the block diagonal matrix whose diagonal blocks arecomposed of matrices M1 , . . . , Mn
R(A, B)the controllable subspace with respect to the pair (A, B)
When the dimensions are clear from the context, we omit the superscript and thesubscript, e.g., employing RH∞ instead of RHn×m∞ . The L2 -norm of a square-integrable
function f ∈ L2 is defined bysZ∞∥f (t)∥2 dt∥f ∥2 :=
0

7
and the H2 -norm and H∞ -norm of a transfer matrix G are defined bys∥G∥2 :=12π

Z ∞
tr (G(jω)G(jω)T ) dω,∥G∥∞ := sup ∥G(jω)∥,ω∈R−∞respectively. The lower and the upper linear fractional transformations (LFTs) [27] aredefined by
Fl (M, N ) := M11 + M12 N (I − M22 N )−1 M21 ,Fu (M, N ) := M22 + M21 N (I − M11 N )−1 M12 ,respectively, where M11 , M12 , M21 , M22 are defined by"M=

M11 M12
M21 M22#whose dimensions are compatible with N , provided that the inverses exist. When atransfer function G has a realization C(sI − A)−1 B + D, the shorthand is defined by
"

A B
C D

#
:= G.

In this thesis, all physical quantities such as the voltage V , are represented in per unit
and the unit is omitted when it is clear from the context. The superscript ⋆ represents
setpoints; for example, P ⋆ denotes the setpoint of P .

Chapter 2
Distributed Design of Local
Controllers via Retrofit Control
2.1

Introduction

In this chapter, a distributed design strategy of local controllers via retrofit control
is proposed. We consider a stably operated network system where there are multiple
operators whose aim is to suppress effects of disturbance injected into the network
system by implementing subcontrollers while preserving the stability. We stand in
an individual operator’s point and treat the network system as an interconnected
system that consists of the subsystems of interest and an unknown environment. The
prior information that the entire system is stable leads to the assumption that theenvironment belongs to the set of systems with which the entire network system isinternally stable. We give a new definition of retrofit controllers as controllers that
preserve the stability of the entire network system for any possible environment andthen derive a necessary and sufficient condition for retrofit controllers based on Youlaparameterization. Next, we introduce two kinds of retrofit controllers, output-rectifying
retrofit controllers and input-rectifying retrofit controllers. We show that a certain
clear structure can be observed inside all output-rectifying retrofit controllers and dual
results can be obtained for input-rectifying retrofit controllers.
This chapter is organized as follows. In Section 2.2, we give the system model
treated in this chapter and the definition of retrofit controllers. Section 2.3 derivesa characterization of all retrofit controllers in the frequency domain based on Youlaparameterization. In Section 2.4, we introduce the notion of output-rectifying retrofit
controllers and input-rectifying retrofit controllers and identify the structure that must
be inside all output-rectifying retrofit controllers with interconnection signal. Moreover,10
in Section 2.5, we overview dual results found in input-rectifying retrofit controllers.
Finally, Section 2.6 concludes this chapter.

2.2

Retrofit Control

2.2.1

System Model

We consider linear time-invariant interconnected systems illustrated by Fig. 2.1, where
the system is given by
 

 
w
v



  z  = G d ,


 
(2.1)
Gpre :

y
u



v
= GE w,v, w, are interconnection signals, z is the evalulated output, d is the external disturbance,
y is the measurement output, u is the control input, G, GE are transfer matrices of thesubsystems, G is given by
Gwv Gwd Gwu
G =  Gzv Gzd Gzu  .
Gyv Gyd Gyu
For ease of discussion, we assume that well-posedness is guaranteed for any feedback
system in this thesis. We refer to G, GE and Gpre as a local system, an environment, anda preexisting system, respectively. The local system is connected to the environment
through interconnection signals and these systems form the preexisting system. In thesetting, the local system represents a subsystem managed by a single operator whereas
the environment includes the other subsystems and also the other sub-controllers
implemented by the other operators. Suppose that the interconnected system stably
operates and our aim is to design a dynamic controller K that generates the feedback
controlu = Ky(2.2)
to improve control performance, namely, L2 -norm of the transfer function from d toz, while preserving the entire system stability. The signal-flow diagram of the entire
system is shown in Fig. 2.2.

11

Fig. 2.1 The signal-flow diagram of the interconnected system between the local system
G and an environment GE .

2.2.2

Definition of Retrofit Controllers

When we can use the model of the system of (2.1) for design of the controller, many
controller design methods such as H∞ optimal control can be utilized. It is, however,difficult to grasp the model information of the whole system when the system represents
a large-scale network system where there are multiple managers such as power systems.
The environment possibly varies due to others’ control action even if the designer canobtain the model information of the whole system at some time instant. This factindicates that it is impossible to follow all variations of the network system undercontrol by multiple operators. Following the circumstance, we suppose that the entire
model of the network system is unavailable. Instead, we assume that only the model
information of G is available and the environment GE can be any transfer matrix
in compliance with keeping the preexisting system to be stable. In other words, we
suppose that GE can be taken as any element of the set GE whereGE := {GE ∈ RP : Gpre is internally stable}.
We refer to a notion of control under the restriction on system model information
as retrofit control and controllers that guarantee the entire system stability for any
possible environment as retrofit controllers. The precise definition of retrofit controllers
is given as follows.
Definition 1 A controller K in (2.2) is called a retrofit controller with respect to G,
when the system in Fig. 2.2 is internally stable for any GE ∈ GE .
One may think that the definition of retrofit controllers can be treated as a kindof robust stabilizing controllers in the sense that retrofit controllers guarantee the12Fig. 2.2 The signal-flow diagram of the closed-loop system.
internal stability of the whole system for any possible unknown perturbation. Retrofit
controllers are, however, essentially different from conventional robust controllers since
the set of perturbations is characterized by the systems that make the preexisting
open-loop system internally stable. On the other hand, in the setting of typical robust
control, e.g., H∞ control, the perturbation is represented by the set of plants whose
norm is bounded.
In the following part, we develop a retrofit control for distributed design. We derivea mathematical characterization of the retrofit controllers and a parameterization of all
retrofit controllers with a constrained design parameter. We also show that the local
closed-loop map from d to z can be represented with affine functions in terms of thedesign parameter of retrofit controllers.

2.3

Characterization of All Retrofit Controllers

In the previous section, the definition of retrofit controllers is given. In this section, we
derive a necessary and sufficient condition for retrofit controllers. Before proceeding to
the main theorem, we note that the set of environments can be characterized based on
the Youla parameterization [28]. For the sake of simplicity, the following assumption ismade throughout this chapter:
Assumption 1 The local system is stable; that is, G belongs to RH∞ .
Under Assumption 1, the Youla parameterization with respect to GE has a simple
form as follows. For any GE ∈ GE , there exists QE ∈ RH∞ such that GE = (I +13
QE Gwv )−1 QE and conversely for any QE ∈ RH∞ there exists GE ∈ GE satisfying
QE = (I − GE Gwv )−1 GE . Thus the set GE can be characterized by
GE = {(I + QE Gwv )−1 QE ∈ RP : QE ∈ RH∞ }.
We hereafter refer to QE as the Youla parameter with respect to Gwv . Note that thisassumption is not very essential and the extended results are shown in Chapter 4.

2.3.1

Characterization of All Retrofit Controllers

A characterization of all retrofit controllers is given by the following theorem.
Theorem 1 Consider the interconnected system in Fig. 2.2. Under Assumption 1,
the controller K is a retrofit controller if and only if there exists Q ∈ RH∞ such that
K = Q(I + Gyu Q)−1

(2.3)

Gwu QGyv = 0.

(2.4)

and

Proof: (sufficiency) We first prove the sufficiency that if there exists Q ∈ RH∞
such that the condition (2.4) holds then K is a retrofit controller. Note that Q is the
Youla parameter with respect to Gyu given by
Q = (I − KGyu )−1 K.
Because for any GE in GE , K = 0 achieves the internal stability of the entire system,
the system in Fig. 2.2 is stabilizable with respect to K for any GE in GE . Hence theinternal stability of the system in Fig. 2.2 is equivalent to that of the system in Fig. 2.3
from Lemma 12.2 in [27]. The internal stability of the system is equivalent to that the
sixteen transfer matrices from δu , δy , δv , δw to u, y, v, w belong to RH∞ .
First, we focus on the signal w. From the block diagram in Fig. 2.3,
w = (Gwu QGyv GE + Gwv GE )w + Gwu (I − KGyu )−1 δu + Gwu Qδy + Gwv δv + δw .
The condition (2.4) implies thatw = (I − Gwv GE )−1 Gwu (I − KGyu )−1 δu + (I − Gwv GE )−1 Gwu Qδy+(I − Gwv GE )−1 Gwv δv + (I − Gwv GE )−1 δw .

(2.5)

14

Fig. 2.3 An equivalent system to that in Fig. 2.2 for stability analysis.
Using the relations(I − Gwv GE )−1 = I + Gwv QE ,(I − KGyu )−1 = I + QGyu ,the relationship (2.5) can be transformed intow = (I +Gwv QE )Gwu (I +QGyu )δu +(I +Gwv QE )Gqu Qδy +QE δv +(I +Gwv QE )δw . (2.6)
Because Q and QE belong to RH∞ , the stability of G implies that the transfer matricesin (2.6) belong to RH∞ . With (2.6), we havev = QE Gwu (I + QGyu )δu + QE Gwu Qδy + (I + QE Gwv )δv + QE δw .
This fact results in that the transfer matrices from δu , δy , δv , δw to v, w belong to RH∞ .
Hence, the relationship(u = (I + QGyu )δu + Qδy + QGyv v,
y = (I + Gyu Q)Gyu δu + (I + Gyu Q)δy + (I + Gyu Q)Gyv v

implies that all of the transfer matrices from δu , δy , δv , δw to u, y, v, w belong to RH∞ .
Therefore, the system in Fig. 2.3 is internally stable for any GE in GE , that is, K is aretrofit controller if Q belongs to RH∞ and the condition (2.4) is satisfied.
(necessity) We next show the necessity that if K is a retrofit controller then Q
belongs to RH∞ and the condition (2.4) is satisfied.

15
First, we take GE = 0 ∈ GE . Since K is a retrofit controller, the system in Fig. 2.3
is internally stable in this case. Then the relationship from δu and δu to u and y canbe written by(u = (I − KGyu )−1 δu + Qδy ,y = (I − Gyu K)−1 Gyu δu + (I − Gyu K)−1 δy .
Due to the internal stability, the matrix from δy to u necessarily belongs to RH∞ andthis leads to that Q belongs to RH∞ .
We next show that the condition (2.4) is satisfied. Define Tloop := Gwu QGyv . Thenthe relation from δw to v is given by (I − QE Tloop )−1 QE . The internal stability implies
that(I − QE Tloop )−1 QE ∈ RH∞ .
(2.7)
Now consider (I − QE Tloop )−1 and then(I − QE Tloop )−1 = I + (I − QE Tloop )−1 QE Tloop ∈ RH∞
holds because (2.7) holds and Tloop belongs to RH∞ .
Finally, supposing
Tloop ̸= 0

(2.8)

we construct an environment GE ∈ GE that destabilizes (I − QE Tloop )−1 . Under thecondition (2.8), there exists QE in RH∞ such thatdet(I − QE (jω)Tloop (jω)) = 0
for some ω ∈ R as in the proof of the small-gain theorem (for example, see Theorem 9.1
in [27]). For such QE , the transfer matrix (I − QE Tloop )−1 does not belong to RH∞ .
This is a contradiction and Tloop must be zero. Therefore, the condition (2.4) holds if
the system in Fig. 2.3 is internally stable for any GE ∈ GE .
2
Theorem 1 gives a constrained version of the Youla parameterization of all stabilizing
controllers that can cope with the unknown environment. An intuitive interpretationof the claim of Theorem 1 can be explained as follows. Omitting d and z, we havethe closed-loop system depicted by the left part of Fig. 2.4. Based on the Youlaparameterization, we can obtain an equivalent system shown in the right part of
Fig. 2.4. The condition (2.4) indicates that the loop transfer matrix becomes a zeromatrix and hence the closed-loop system is internally stable for any environment. On
the other hand, if the condition (2.4) is not satisfied, then there exists QE such that16

: arbitrary in

Fig. 2.4 Equivalent closed-loop systems
the system becomes unstable because QE can be chosen as any element of RH∞ fromthe definition of GE . Thus, the condition (2.4) is also a necessary condition for retrofit
controllers.

2.3.2

Closed-Loop Map with Retrofit Controller and Difficulty
on Retrofit Controller Design

The closed-loop map of the entire system with a retrofit controller is given by thefollowing theorem.
Theorem 2 Let Tall be the closed-loop map from the external disturbance d to theevaluated output z. If K is a retrofit controller, thenTall = Tzd (Q) + Tzv (Q)QE Twd (Q)where
Tzd (Q) := Gzd + Gzu QGyd ,Tzv (Q) := Gzv + Gzu QGyv ,Twd (Q) := Gwd + Gwu QGyd .

(2.9)

17
Proof: By definition, we have
Tall = Fl (Gpre , K)
= Fl (Fu (G, GE ), K)= Fu (F"l (G, K), GE ) #= Fu

Gwv Gwd
Gzv Gzd

"
+

Gwu
Gzu

#

!
Q[Gwv Gyd ], GE= Gzd + Gzu QGyd+(Gzv + Gzu QGyv )GE {I − (Gwv + Gwu QGyv )GE }−1 )(Gwd + Gwu QGyd ).
From the assumption that K is a retrofit controller and Theorem 1, Gwu QGyv = 0 andhence Tall can be written as (2.9).
2The closed-loop map with a retrofit controller is an affine function of QE and transfer
matrices Tzd , Tzv , Twd are also affine functions of Q while this property is not satisfied
when a general controller is employed. From Theorem 2, it is expected that the entire
closed-loop performance is improved by simply choosing Q so as to suppress the affine
transfer matrices. Note that those transfer matrices represent the local closed-loop
maps composed of G and K from the external input and the interconnection signal
from the environment to the evaluated output and the interconnection signal from thelocal system. Consequently, it is indicated that we have to take not only d and z but
also v and w into account for control performance. Conversely, it is also shown thatthe only alternative to deal with the varying environment is to simultaneously suppress
the transfer matrices.
It has been revealed that the retrofit controller design problem is reduced to the
problem to find Q ∈ RH∞ such that norms of the transfer matrices in (2.9) are small
under the constraint (2.4). The design problem, however, cannot be handled by astandard controller design method even though the optimization problem is convex
owing to the existence of the condition (2.4). One approach to overcome the difficulty
is to provide an explicit representation of all solutions to the equation (2.4) but thisapproach is severe in general. In fact, all solutions to the equation (2.4) in R can be
represented by the form
Q = Q0 − G†wu Gwu Q0 Gyv G†yv ∈ R(2.10)

where G†wu and G†yv are the generalized inverses of Gwu and Gyv , respectively, and Q0
is an arbitrary transfer matrix provided that Gwu and Gyv are full-row and full-column

18rank (See Fact 6.4.43 in [29]). However, it is difficult to design a retrofit controller bydirectly utilizing (2.10) for the following two reasons. First, solutions derived by (2.10)
are not guaranteed to be proper and stable because the generalized inverses are possibly
improper or unstable. Second, the design parameter in (2.10) is represented by Q0 andthe transfer matrices in (2.9) are no longer affine functions with respect to the design
parameter.
The above discussion indicates that retrofit controller design is a hard problem tosolve in the general case. To resolve this issue, in the next section we consider twoclasses of retrofit controllers and derive explicit representations for the classes of retrofit
controllers with which the design problem is reduced to be a standard problem.

2.4

Design of Output-Rectifying Retrofit Controllers
with Interconnection Signal

In this section, we consider two classes of retrofit controllers and show that the retrofit
controllers of the classes can be parameterized with a local stabilizing controller. It has
been already shown that a retrofit controller can be designed by letting the controller
have a structure shown below in [19]. We reveal that the structure is not only sufficient
but also necessary for a class of retrofit controllers.

2.4.1

Definition of Output-Rectifying Retrofit Controllers andInput-Rectifying Retrofit Controllers

Now we define two kinds of retrofit controllers, namely, output-rectifying retrofit
controllers and input-rectifying retrofit controllers.
Definition 2 A controller K is said to be an output-rectifying retrofit controller, if
there exists Q ∈ RH∞ such that (2.3) holds and
QGyv = 0.

(2.11)

Similarly, a controller K is said to be an input-rectifying retrofit controller, if there
exists Q ∈ RH∞ such that (2.3) holds and
Gwu Q = 0.

(2.12)

19
Obviously, those controllers are retrofit controllers because the conditions (2.11)
and (2.12) are sufficient conditions for the condition (2.4). The reason why thecontrollers satisfying (2.11) are referred to as output-rectifying retrofit controllers isthat the measurement output is rectified inside the controller such that the effectfrom the interconnection signal from the environment v is eliminated. Similarly, thecontrollers satisfying (2.12) are referred to as input-rectifying retrofit controllers because
the control input is rectified such that the input does not affect the interconnection
signal to the environment w at all.

2.4.2

Parameterization of All Output-Rectifying Retrofit Controllers with Interconnection Signal

We attempt to provide an unconstrained parameterization of all output-rectifying
retrofit controllers. We first consider the special case where the interconnection signal
v is available at the controller; that is, the control input can be represented by
"
u=K

y
v

#
.

(2.13)

It is shown that all output-rectifying retrofit controllers can be parameterized with alocal stabilizing controller under the measurability of the interconnection signal. We also
show that all output-rectifying retrofit controllers with interconnection measurement
have a specific structure composed of a local staibilizing controller and an output
rectifier, defined below. This case under the assumption that the interconnection signal
can be fed back has not only advantage to simplify the results but also implication for
more general cases.
The following theorem gives an unconstrained parameterization of all outputrectifying retrofit controllers with interconnection measurement.
Theorem 3 Under the assumption that the interconnection signal v can be fed backand K can be represented as (2.13), the controller K is an output-rectifying retrofit
controller if and only if the controller has the form
K = K̂GR
where K̂ is a controller that stabilizes Gyu with
GR := [I − Gyv ].

(2.14)

20
Proof: First of all, we define the transfer matrices G(y,v)v , G(y,v)d , and G(y.v)u by
"

y
v

#




v

 
= G(y,v)v G(y,v)d G(y.v)u  d  .
uSpecifically, those transfer matrices are given by"G(y,v)v =

Gyv
I

#

"
, G(y,v)d =

Gyd
0

#

"
, G(y,v)u =

Gyv
0

#
.

Note that the condition for output-rectifying retrofit controllers (2.11) is rewritten by
QG(y,v)v = 0(2.15)

with Q := (I − KG(y,v)u )−1 K under the assumption that v can be fed back.
Assume that Q ∈ RH∞ and the condition (2.15) is satisfied. Suppose K can bedivided byhi
K = K̂ K̂v
with transfer matrices K̂ and K̂v whose dimensions are compatible with y and v,
respectively. Note that the condition (2.15) is equivalent tohiK̂ K̂v G(y,v)v = 0

that implies
K̂v = −K̂Gyv .
Hence, the controller has the form (2.14). We now show that K̂ is a stabilizing controller
for Gyu . Consider
Q̂ := (I − K̂Gyu )−1 K̂and note that

Q = (I − KG(y,v)u )−1 K= (I − K̂Gyu )−1 K̂R= Q̂[I − Gyv ].

(2.16)

By multiplying [I 0]T to (2.16) from the right side we have Q̂ = Q[I 0]T . From theassumption that Q ∈ RH∞ , Q̂ belongs to RH∞ as well. This leads to that K̂ stabilizes
Gyu .

21

Fig. 2.5 Structure inside the output-rectifying retrofit controllers with the interconnection signal.
To prove the converse, we assume that the controller can be represented by (2.14)
and K̂ is a stabilizing controller for Gyu . Simple calculation leads to the condition (2.15)
and the relationship (2.16) implies that Q belongs to RH∞ . Therefore, K is an outputrectifying retrofit controller.
2
Theorem 3 indicates that all output-rectifying retrofit controllers with the interconnection signal from the environment hold the structure depicted by Fig. 2.5. If we
implement K̂ alone as a controller K, it is obvious that the entire system stability
cannot be guaranteed since the measurement signal y is influenced by the interconnection signal v from the environment GE . The transfer matrix Gyv in Fig. 2.5 has
the role to simulate the effect of the interconnection signal to the measurement signal.
Theorem 3 also implies that by placing the output rectifier GR inside the retrofit
controller we can produce the rectified measurement signal y − Gyv v = GR [y T v T ]T andstabilize the entire interconnected system using K̂ and also this procedure is the onlymeans to cope with all possible environments. Furthermore, Theorem 3 claims that
all output-rectifying retrofit controllers can be parameterized with a local stabilizing
controller K̂ when the interconnection signal can be fed back. Thus the new design
parameter Q̂ can be taken as any element in RH∞ . It should be emphasized that thestructure in Fig. 2.5 is the exactly same as that introduced in the existing study [19], inwhich it is shown that controllers that have the structure become a retrofit controller,that is, the structure is a sufficient condition for retrofit control.
To discuss control performance, we explicitly give the closed-loop map when the controller is an output-rectifying retrofit controller with the interconnection measurement.
The following theorem holds.

22
Theorem 4 If K is an output-rectifying retrofit controller with the interconnection
signal, then the closed-loop map from d to z can be represented by
Tall = T̂zd (Q̂) + Gzv QE T̂wd (Q̂),
whereT̂zd (Q̂) := Gzd + Gzu Q̂Gyd ,T̂wd (Q̂) := Gwd + Gwu Q̂Gyd .

Proof: From Theorem 2, the closed-loop map can be represented by (2.9). Since"#
Gyd
QG(y,v)d = Q
0
"
#
Gyd
= Q̂GR
0= Q̂Gyd ,we have Tzd (Q) = T̂zd (Q̂) and Twu (Q̂) = T̂wu (Q̂). Moreover, the condition (2.15) leads
to that Tzv (Q) = Gzv . Substituting these relationships to (2.9) yields the claim.
2
Theorem 4 implies that not only the closed-loop map is an affine function withrespect to QE but also the transfer matrices T̂zd and T̂wu are affine functions withrespect to the design parameter Q̂ as well. It should be emphasized that the design
parameter in Theorem 4 is represented by Q̂ that is unconstrained in contrast to
the case of Theorem 2. Theorem 4 also implies that the effects from the external
disturbance d to the evaluated output z and the interconnection signal w should be
suppressed simultaneously when the model information on QE is unknown. In summary,the result suggests that for a retrofit controller design problem we only have to solve
an unconstrained convex optimization problem where the design parameter is Q̂ thatsuppresses both T̂zd and T̂wd and construct the associated output-rectifying retrofit
controller according to the structure in Fig. 2.5 using the corresponding K̂. The block
diagram of the closed-loop map is illustrated by Fig. 2.6.

23

Fig. 2.6 Block diagram of the closed-loop map with an output-rectifying retrofit
controller.

2.5

Dual Results in Input-Rectifying Retrofit Controllers

In this subsection, we focus on input-rectifying retrofit controllers and derive dual
results that correspond to the theorems shown in the previous section. It is revealed
that all input-rectifying retrofit controllers can be parameterized with an unconstrained
design parameter under dual assumptions as with the case of output-rectifying retrofit
controllers.
While we have considered the case in which the interconnection signal v is available
at K, we consider the case where control input can be directly injected into theinterconnection signal w as a dual case; that is, the control input can be represented by
"

u
uw

#
= Ky

(2.17)and the interconnection signal w can be represented by
w = Gwv v + Gwd d + Gwu u + uw .

(2.18)

It is shown that all input rectifying retrofit controllers can be parameterized with alocal stabilizing controller and have a dual structure under the assumption as with
output-rectifying retrofit controllers.
The following theorem holds.

24
Theorem 5 Assume that the control input and the interconnection signal can be
represented by (2.17) and (2.18), respectively. Then K is an input-rectifying retrofit
controller if and only if the controller has the form
K = GdR K̂,
where K̂ is a controller that stabilizes Gyu with"
GdR :=

I
−Gwu

#
.

Furthermore, the closed-loop map with an input-rectifying retrofit controller can be
represented byddTall = T̂zd(Q̂) + T̂zv
(Q̂)QE Gwd ,
where
d
T̂zd
(Q̂) := Gzd + Gzu Q̂Gyd ,dT̂zv(Q̂) := Gzv + Gzu Q̂Gyv ,with Q̂ := K̂(I − Gyu K̂)−1 .
Proof: Assume that Q ∈ RH∞ and the condition (2.12) is satisfied. Suppose
"

u
uw

#

"
=

K̂
K̂w

#
y,

with transfer matrices K̂ and K̂w . Note that the condition (2.12) is written by"[Gwu I]

K̂
K̂w

#
= 0,

in this case. Then the condition (2.12) implies thatK̂w = −Gwu K̂,and hence the controller can be represented by (2.17). Now we show that K̂ is astabilizing controller for Gyu . Consider Q̂ := K̂(I − Gyu K̂)−1 and
Q = GdR Q̂.

25

Fig. 2.7 Structure inside the input-rectifying retrofit controllers under full-control forthe interconnection signal.

Fig. 2.8 Block diagram of the closed-loop map with an input-rectifying retrofit controller.
By multiplying [I 0] from the left side we obtain the conclusion. Finally, simple
calculation leads to the closed-loop map as with Theorem 4.
2
Theorem 5 indicates that all input-rectifying retrofit controllers under the assumption hold the structure depicted by Fig. 2.7. The structure can be regarded as adual one in Fig. 2.5. While the output rectifier in Fig. 2.5 has the role to rectify themeasurement output, the input rectifier GdR has the role to rectify the control input.
According to Theorem 5, as with the case of the output rectifying retrofit controllers,
the structure in Fig. 2.7 is a necessary and sufficient condition for guaranteeing thestability for any possible environment. Moreover, the degree of freedom of the controller
design is expressed by K̂ that stabilizes Gyu in this case as well. The block diagram of
the closed-loop map is illustrated by Fig. 2.8.

26

2.6

Chapter Summary

In this chapter, we have proposed a novel retrofit control approach for distributed
design of local controllers. We have first introduced the definition of retrofit controllers
and derived a necessary and sufficient condition to characterize retrofit controllers
in the frequency domain using Youla parameterization. It has been seen that theretrofit controller design problem is rather difficult in the general situation. To resolve
this issue, two classes of retrofit controllers, output-rectifying retrofit controllers andinput-rectifying retrofit controllers, have been defined. It has been reveled that all
output-rectifying retrofit controllers can be parameterized under the assumption thatthe interconnection signal from the environment can be fed back and its design problem
is much easier than the general case. Moreover, dual results in input-rectifying retrofit
controllers have been shown. These results enable us to design local controllers for alarge-scale dynamical network system in a distributed manner.
Although the results establish a fundamental theory for distributed design based on
retrofit control, there remain several issues to be resolved. First, the assumption on themeasurability of the interconnection signal is required in theory but this assumption is
not always satisfied. The second issue is that the effectiveness has not been verified inpractical systems yet. Finally, the information structure of the designed controllers are
purely decentralized. The aim of the following chapters is to resolve these issues.

Chapter 3
Output-Rectifying Retrofit
Controllers without Interconnection
Signal3.1

Introduction

Our aim in this chapter is to develop a design method of output-rectifying retrofit
controllers without interconnection signal. In practical network systems, it is desirable
to design a retrofit controller without interconnection signal because implementing
sensors to obtain the interconnection signal often costs and it is sometimes impossible
to measure the interconnection signal. For ease of discussion, we consider state-feedback
output-rectifying retrofit controllers and show that a similar parameterization to thatin the previous chapter is available in this case as well.
This chapter is organized as follows. In Section 3.2, we provide the system model
treated in this chapter and consider the design problem of a state-feedback outputrectifying retrofit controller. Section 3.3 gives a parameterization of all state-feedback
output-rectifying retrofit controllers and proposes a design method. Finally, Section 3.5
concludes this chapter.

28

3.2

System ModelIn this chapter, we consider linear time-invariant interconnected systems (2.1) andassume that the state-space representation of G is given byẋ = Ax + Lv + Rd + Bu
 w = ΓxG:z = Sx



y =x

(3.1)

where x is the state of G. Without loss of generality, we assume that L and B are
full-column rank matrices. In the state-space representation, feedthrough terms areomitted for simplicity. Our purpose is to suppress the effect from d to z by designing a
retrofit controller K for
u = Kx(3.2)
without the interconnection signal v.
In the previous chapter, we have observed that all output-rectifying retrofit controllers can be represented with an unconstrained design parameter when the interconnection signal is measurable. In this chapter we attempt to relax this assumption.
It will be shown that all state-feedback output rectifying retrofit controllers have astructure similar to that in Fig. 2.5.

3.3

Parameterization of All State-Feedback OutputRectifying Retrofit Controllers

This section gives a parameterization of all state-feedback output-rectifying retrofit
controllers. The basic idea for synthesizing a state-feedback retrofit controller is
explained as follows [19]. For P being a full-row rank matrix such that
im L ⊂ ker P,
we consider applying the basis transformation of†

x = P †ϕ + P ϕ
with
ϕ := P x,

ϕ := P x

(3.3)

29
†

where P is a matrix that satisfies the relationship P † P + P P = I. Then the dynamics
with respect to ϕ can be described as†ϕ̇ = P AP † ϕ + P AP ϕ + P Lv + P Rd + P Bu†

= P AP † ϕ + P AP ϕ + P Rd + P Bu.

(3.4)

The key observation here is that L is eliminated in (3.4) owing to the condition (3.3).
By regarding ϕ and ϕ as a new measurement signal and a virtual interconnection signal,
respectively, we can construct a retrofit controller for the reduced-order model (3.3)
based on the structure in Chapter 2. Note that the above procedure gives a sufficient
structure of state-feedback retrofit controllers, but does not claim any necessity of thecontroller structure.
The following theorem shows that all state-feedback output-rectifying retrofit controllers without interconnection signal are necessarily synthesized by such a procedure.
Theorem 6 The controller K in (3.2) is a state-feedback output-rectifying retrofit
controller without interconnection signal if and only if the controller can be represented
by
K = K̂ G̃R(3.5)
where K̂ is a controller that stabilizes
Gϕu := (sI − P AP † )−1 P B
and P is a full-row rank matrix such that the condition(3.6)

im L = ker P
holds with†G̃R := P − (sI − P AP † )−1 P AP P .
Proof: The proof is built on two parts. In the former part, we prove that all retrofit
controllers have the form (3.5). In the later part, we prove that K̂ belongs to the classof controllers that stabilize Gϕu .
In the former part, we show that all solutions in RP to the equation (2.11) isrepresented as the form (3.5), or equivalently,
KGyv = 0 ⇔ ∃ K̂ ∈ RP s.t. K = K̂ G̃R .

(3.7)

30
Note that Gyu = (sI − A)−1 L in this chapter.
As a preparation, we give all solutions in R to the equation (2.11). Now let nbe the dimensions of x. Because R is a vector space, the fundamental theorem onhomomorphisms leads to that for K ∈ R
KGyv = 0 ⇔ ∃ K̂ ∈ R s.t. K = K̂Ξ(3.8)

for any transfer matrix Ξ that is an element in R and satisfies the conditions
dim(im Ξ) = n − dim(im Gyv )(3.9)

ker Ξ = im Gyv ⊂ Rn .

(3.10)

Ξ := P {I − (sI − A)−1 LL† (sI − A)} ∈ R.

(3.11)

andWe now takeLet us confirm that the above transfer matrix satisfies the conditions. By checking thefeedthrough term of Ξ, we havelim Ξ(s) = P (I − LL† )s→∞and Ξ ∈ RP. Because

im (I − LL† ) = (im L)⊥= (ker P )⊥from the condition (3.6), P (I − LL† ) is full-row rank. Since Ξ belongs to RP and its
feedthrough term is right-invertible, Ξ is right-invertible in RP. Hence,dim(im Ξ) = dim(im P )and Ξ satisfies the condition (3.9). Moreover, we haveΞGyv = P {Gyv − (sI − A)−1 L}=0and thus Ξ satisfies the condition (3.10) as well. The above discussion implies that, forsuch Ξ, if K ∈ R satisfies the condition in (3.8), there always exists K̂ ∈ R such thatthe condition in (3.8) is satisfied, and the converse holds as well.

31
Next, we show that K satisfying the condition in (3.8) for Ξ defined by (3.11)
belongs to RP if and only if K̂ belongs to RP, or roughly, K is proper if and only if
K̂ is proper. From the relationship K = K̂Ξ, it is obvious that if K̂ is proper then
K is proper. For the converse, we have KΞ† = K̂ by multiplying Ξ† . Because Ξ† also
belongs to RP, if K is proper then K̂ is proper. Thus we see that for K ∈ RP
KGyv = 0 ⇔ ∃ K̂ ∈ RP s.t. K = K̂Ξ.
We finally show that Ξ = G̃R . We have−1−1†
Ξ = P [I − (sI# " − A) Ls} ]#!
" − A) Ls{(sIA − ALL† ALL†
A AL
=P I−
I L
−L†
L†

because

"

A B
C D

#†

"
=

A − BD† C BD†
−D† C
D†

#

provided that D is left-invertible. Since"

we have

A2 B2
C2 D2

#"

A1 B1
C1 D1

#




A1
0
B1


=  B2 C1 A2 B2 D1  ,D2 C1 C2 D2 D1


†
ALL
0 ALL†
Ξ = P I −  −ALL† A ALL†  .
−LL† I LL†

By considering the coordinate transformation
"

I −I
0 I

#

32
we obtain
 

†
0
A
0
ALL
0 ALL†† ALL† 
 −ALL† A ALL†  =  −ALL† ALL−LL† I LL†−LL† I − LL† LL†#"†
ALL ALL†
.
=
†
LL
LL†


Again, considering the coordinate transformation"

we have
"

†

ALL

ALL

†

LL†

LL

and hence

†

#

P
P

#
,


0 P AP † P ALL†
=  0 P AP † P ALL† 
L
LL†#
" 0
P AP † P ALL†
=
L
LL†

"



†

P AP † P AP PΞ=−IP†= G̃R .

#

Therefore, it turns out that (3.7) holds.
In the later part, we show that K is a retrofit controller if and only if K̂ in (3.7)
belongs to the class of controllers that stabilize Gϕu . For simplicity of discussion, we
assume thatGϕu ∈ RH∞ .
Then what we have to show is written as
Q ∈ RH∞ ⇔ Q̂ ∈ RH∞
where Q̂ := (I − K̂Gϕu )−1 K̂.

33
Now we have
#
#"
†
A B
P AP † P AP P
G̃R Gyu =
I 0
−I
P†


A
0
B
†


=  P AP P P AP † 0 
P†
−I
0


B
A
0


†
=  0 P AP −P B 
−I
" 0
#0
P AP † P B
=
I
0
= Gϕu
"

with the coordinate transformation"

Therefore

I 0
−P I

#
.

Q = (I − K̂ G̃R Gyu )−1 K̂ G̃R= (I − K̂Gϕu )−1 K̂ G̃R= Q̂G̃R .

From the assumption that Gϕu ∈ RH∞ , G̃R also belongs to RH∞ . Thus if Q̂ ∈ RH∞
then Q ∈ RH∞ . Furthermore, since G̃R is right-invertible, we have
QG̃†R = Q̂.
Because"G̃†R =†A P P AP †IP†G̃†R ∈ RH∞ . Hence, if Q ∈ RH∞ then Q̂ ∈ RH∞ .

#,2.

Theorem 6 implies that all state-feedback output-rectifying retrofit controllers canbe parameterized with K̂ that stabilizes the reduced-order model Gϕu and have the

34Fig. 3.1 Structure inside the state-feedback output-rectifying retrofit controllers whereGϕϕ is defined by (3.16).
structure depicted by Fig. 3.1 with†Gϕϕ := (sI − P AP † )−1 P AP .

(3.12)It should be noted that the same structure in Fig. 2.5 can be observed in Fig. 3.1.
Specifically, y, v and Gyv in Fig. 2.5 correspond to ϕ, ϕ, and Gϕϕ , respectively. Notealso that the proof cannot be directly extended to output feedback retrofit controllers
because it is difficult to find a basis of (im Gyv )⊥ in the output feedback setting while
G̃R is the basis for the state-feedback case.
The closed-loop map from d to z with a state-feedback retrofit controller can be
represented with an unconstrained design parameter as well. The following theorem
holds.
Theorem 7 If K is a state-feedback output-rectifying retrofit controller, then theclosed-loop map from d to z can be represented by
Tall = T̃zd (Q̂) + Gzv QE T̃wu (Q̂)
whereT̃zd (Q̂) := Gzd + Gzu Q̂Ĝxd ,T̃wu (Q̂) := Gwd + Gwu Q̂Ĝxd

withĜxd := (sI − P AP † )−1 P R,

Q̂ := (I − K̂Gϕu )−1 K̂.

Proof: Because
Gϕu = G̃R Gxd ,Ĝxd = ĜR Gxd ,35
we have

QGxd = (I − K̂ G̃R Gxu )−1 K̂ G̃R Gxd= Q̂Ĝxd .

Therefore, as with the proof of Theorem 3, the claim holds.

2

The transfer matrices T̃zd and T̃wu in Theorem 7 are affine functions with respect
to the free design parameter Q̂ as in Theorem 3. Thus the design problem of statefeedback output-rectifying retrofit controllers can be treated as a standard unconstrained
controller design problem.

3.4

Dual Results

In this section, we give dual results on input-rectifying retrofit controllers. The logic
for the statements in this subsection is the same as that in the previous section.
Consequently, concrete proofs are omitted here.
We consider the following systemẋ = Ax + Lv + Rd + u w = ΓxG:z = Sxy = Cx(3.13)

for the local system instead of (3.1). The difference of (3.13) from (3.1) is that the
input matrix B is equal to the identify and the measurement signal is not the state x
itself but Cx. We place an assumption that Γ and C are full-row rank matrices. The
controller is given byu = Ky.
(3.14)As a dual result of Theorem thmc3:str, a parameterization of input-rectifying
retrofit controllers is given by
K = G̃dR K̂
where K is a controller that stabilizes
Gdyu := CP (sI − P † AP )−1

36

Fig. 3.2 Structure inside input-rectifying retrofit controllers where ψ, ψ, and Gψψ aredefined by (3.15) and (3.16).
and P is a full-column rank matrix such that the condition
ker Γ = im P
holds with†G̃dR := P − P P AP (sI − P † AP )−1
As shown in Fig. 3.2, the dual structure can be observed in this case compared with
Fig. 3.1 where ψ and ψ are defined by
u = Pψ + Pψand†Gψψ := P AP (sI − P † AP )−1 .

(3.15)(3.16)

Furthermore, the closed-loop map with an input-rectifying retrofit controller can be
represented byddTall = T̃zd(Q̂) + T̃zv
(Q̂)QE Gwd
where
d
T̃zd
(Q̂) := Gzd + Ĝzu Q̂Gyd ,dT̃wd(Q̂) := Gwd + Ĝwu Q̂Gyd ,withĜzu := S(sI − P † AP )−1 ,Ĝwu := Γ (sI − P † AP )−1 .

For design of input-rectifying retrofit controllers, the same procedure can be performed
as that for output-rectifying retrofit controllers.

37

3.5

Chapter Summary

In this chapter, we have considered design of retrofit controllers without interconnection
signal and extend the result in Chapter 2. A parameterization of all state-feedback
retrofit controllers has been derived. It has been also shown that a similar structure
can be observed in state-feedback retrofit controllers to that of retrofit controllers withinterconnection signal. Finally, we have seen that dual results can be obtained for
input-rectifying retrofit controllers.

Chapter 4Distributed Design of Power System
Stabilizers in Power Grids withRenewable Energy Resources
4.1

Introduction

In this chapter, we consider the stabilization problem of power grids where a large
amount of renewable energy resources is introduced. Many controller design methods
have been proposed in the control community, but most of them require the entire
system model and this assumption is not realistic as stated in Chapter 1. In contrast,
distributed design has been practically employed for stabilizing controllers, e.g., PSSs.
The typical distributed design method uses the single-machine infinite-bus system [21]that fucuses only on the dynamics of the subsystem of interest and ignores the otherparts. As a result, assurance of the stability of the entire grid is not theoreticallyobtained. To resolve the issue, a novel distributed design method via retrofit control is
proposed. The advantage of the proposed method is that theoretical guarantee of thestability of the entire grid is obtained.
We first review the importance of frequency control of power grids where a large
amount of PV is penetrated. To develop sophisticated power grids, appropriate usageof renewable energy resources including PV plants plays an important role [30]. The
feed-in tariff (FIT) [31] accelerates the progress of installing PV power generators and,as a result, the number of PV power plants in Japan is mounting up. The total quantityof installed PV generation is anticipated to reach 64 GW by 2030. It covers 7% of
the total energy consumption approximately [32]. Due to the large installation of PV

40plants, some thermal power plants have to stop its operation for balancing the supply
and demand since the demand is invariant though the amount of generated energy isincreased by the installed PV plants. In [33–35], it has been pointed out that power
grids tend to become unstable by stopping a number of synchronous generators dueto the poor controllability of PV plants [36]. This poses the stabilization problem for
future grids and a novel design method for PSSs that improve damping performance ofpower grids with PV plants while keeping the stability.
Distributed control [37–39] is a promising approach that fulfills the requirement.
In distributed control, control inputs generated from local controllers depend only on
the corresponding local measurement signal, e.g., deviation of the frequency of theassociated synchronous generator. However, these distributed control methods arebuilt on the premise that the system model of the whole power grid is utilizable forcontroller design. As noted above, since power grids are totally complex, the wholemodel is inaccessible. This fact leads to that these approaches are not very appropriate
for practical power grids.
The aim of this chapter is to develop a distributed design method for PSSs basedon retrofit control and to show how retrofit control works for the task. It is observedthat the proposed method can significantly reduce the effect of disturbance injected
into the power grid through a numerical simulation.
The organization of this chapter is as follows. We give a mathematical model of thepower grids in Section 4.2. The PSS design problem is mathematically formulated in
Section 4.3. Furthermore, a distributed design method for PSSs is proposed via retrofit
control developed in the previous sections. For the purpose of demonstrating theefficiency of the approach, Section 4.4 shows a numerical simulation for PV-integrated
IEEJ EAST 30-machine power system [25], called EAST30 model for short. Section 4.5
concludes this chapter.

4.2

Modeling of PV-Integrated Power Systems

First of all, we give a mathematical model of a power grid consisting of synchronous
generators, loads, PV plants, and transmission lines of EAST30 model [25] in thestate-space form. For giving a reasonable model, neighboring synchronous generators,
loads, and PV plants are treated as aggregated ones. Buses have the role to interconnect
the aggregated facilities. Throughout this thesis, we refer to the buses to which no
facilities are not assigned as non-unit buses, and a facility connected to a bus or a
non-unit bus as a component.

41
Let us suppose that there exist N buses in the power grid. Let N := {1, . . . , N }and for k ∈ N, the mathematical model of the kth component is given by(Σ[k] :

ẋ[k] = f[k] (x[k] , v [k] , i[k] , us,[k] , u[k] )0 = g[k] (x[k] , v [k] , i[k] ),(4.1)

where x[k] is the state, us,[k] is the supplementary control input, u[k] is the control
input from the corresponding PSS, and i[k] and v [k] are the bus current and voltage,
respectively.
The network structure of the grid is given by
i=Yv

(4.2)

where
i := col(i[k] )k∈N ,v := col(v [k] )k∈N

and Y ∈ CN ×N is the admittance matrix. For the specific values of Y in the targeted
power grid, see [40].
The model of the whole power grid can be obtained by combining (4.1) and (4.2).
We give more detailed models of synchronous generators, loads, PV power plants, andnon-unit buses as (4.1) In the following, the index set of the buses connecting to thegenerators, loads, PV power plants, and non-unit buses are defined by NG , NL , NP and
NN , respectively. Those sets are assumed to be disjoint and NG ∪ NL ∪ NP ∪ NN = N
holds.

4.2.1

Synchronous Generator

We give a model of synchronous generators that have the form (4.1). In a typicalcase, a synchronous generator is composed of a synchronous machine, an excitation
system with automatic voltage regulator (AVR), and a turbine with governor [2]. The
excitation system has the role to feed current to the synchronous machine. The turbinewith governor rotates the machine by feeding a mechanical power. The synchronousmachine converts the mechanical power to electrical power and provides the generated
electrical power to the power system. For k ∈ NG , the mathematical model of the42synchronous machine associated with the kth bus is given byδ̇[k] = ω̄ω[k] 





1
1

T

ω̇[k] =
Pm,[k] − v[k] i[k]
M[k] 1 + ω[k]
Ψ[k] = AΨ[k] Ψ[k] + BΨ[k] i[k] + BV [k] Vfd,[k] ,v[k] = CΨ[k] Ψ[k] + DΨ[k] i[k]





v [k] = comp(Tδ[k] v[k] ) i = comp(T −1 i )[k]δ[k] [k]k ∈ NG(4.3)

where δ[k] denotes the rotor angle relative to the frame rotating at the constant reference
speed ω̄ := 100π (rad/sec), ω[k] denotes the rotor angular velocity relative to ω̄, Ψ[k]includes the magnetic flux associated with the field circuit, d-axis damper windings,
q-axis damper windings, Pm,[k] denotes the mechanical power input, Vfd,[k] denotes
the field voltage, and i[k] and v[k] denote the stator current and the stator voltage,
respectively. The inertia constant is represented by M[k] (sec) and Tδ[k] is given by"Tδ[k] :=sin δ[k] cos δ[k]− cos δ[k] sin δ[k]#that is the coordinate transformation matrix from the coordinate system rotating at
each generator’s own frequency to the coordinate system rotating at the rated frequency.
It should be noted that the system (4.3) is related to the coordinate system that rotatesat the rotational speed ω[k] .
The model of the turbine with governor is given by(⋆ζ̇[k] = Aζ[k] ζ[k] + Bζ[k] ω[k] + Rζ[k] (Pm,[k]+ us,[k] )Pm,[k] = Cζ[k] ζ[k]

(4.4)

⋆
where ζ[k] ∈ R4 denotes the state of the turbine with governor, Pm,[k]denotes themechanical power in the steady state, and us,[k] is the supplementary generation control
action under automatic generation control (AGC) [2]. AGC has the role to regulate
frequency to a pre-defined setpoint. The measurement signal of the supplementary
generation control is given by aggregating all frequency deviations [2] and the control
input is given by
Z t X
us,[k] (t) = κs a[k]
ω[k] (τ )dτ
(4.5)
0 k∈N
G

43
where κs is a feedback gain, a[k] is a scaling factor, and the aggregated frequency
Pdeviation k∈NG ω[k] is referred to as an area control error. Moreover, it is supposedthat the parameters in (4.5) are given so as to stabilize the entire system for thesituation u[k] = 0.
The dynamics of the excitation system with AVR is represented as(η̇[k] = Aη[k] η[k] + Bη[k]



⋆
∥v[k]
∥ − ∥v[k] ∥ + u[k]⋆+ Rη[k] Vfd,[k]Vfd,[k] = Cη[k] η[k](4.6)

where η[k] ∈ R3 denotes the state of the excitation system with AVR, v ∗[k] denots thesteady voltage, Vfd,[k] denotes the steady field voltage, and u[k] denotes the control
input from PSS. The excitation system with AVR has the function providing current
to the synchronous machine.
The model of the synchronous generator is given as the combination of (4.3), (4.4),and (4.6). The state of a synchronous generator is defined byx[k] := col(δ[k] , ω[k] , Ψ[k] , ζ[k] , η[k] ) ∈ R13 ,k ∈ NGand f[k] (·, ·) and g[k] (·, ·) obey (4.3), (4.4), and (4.6). Fig. 4.1 depicts the block diagramof the model of the synchronous generator. The measurement output is given byy[k] := col(ω[k] , v[k] )for each generator.
The specific values of the parameters of synchronous generators are given as follows.
There exist three sorts of generator, that is, thermal power plants, nuclear power plants,
and hydroelectric power plants, in the power system considered in this thesis. For
simplicity, the subscript k ∈ NG is omitted. The matrices of the synchronous machinein (4.3) are−(1 + γd ) Kd γd
0
0 

′
′
Tdo
Tdo




1
−1


0
0
′′
′′


Tdo
 Kd TdoAΨ = 
,−(1 + γq ) Kq γq 


0
0


′
′


Tqo
Tqo



1
−1 
0
0
′′
′′
Kq TqoTqo

44Power systemstabilizer

Turbinewith governor

Exciter with
automatic
voltage regulator

SynchronousmachineCoordinatetransformation

Coordinatetransformation

Power gridFig. 4.1 Block diagram of the model of the generator.
′′(Xd − Xd′ )(Xd − Xl )
−

′
Tdo
(Xd′ − Xl )X ′ − Xl− d ′′

Kd TdoBΨ = 


0



0


′
BV = [1/Tdo
0 0 0]T00CΨ =  ′′′′ Xd − Xl
Xd′ − XKdXd′ − Xl
Xd′ − Xl


′′
Xq 
 0
DΨ = 
,
′′
−Xd 0


0




0


′′
,
(Xq − Xq′ )(Xq − Xl ) ′ (X ′ − X )

Tqo
l
q

′
Xq − Xl′′
Kq Tqo′′′′
Xq − Xl
Xq′ − XqKq ′
Xq′ − Xl
Xq − Xl 
,

0
0

45

′′

(X◦ − X◦′ )(X◦′ − X◦ )γ◦ :=,(X◦′ − Xl )2′′(X◦′ − Xl )(X◦ − Xl ),K◦ := 1 +(X◦′ − X◦′′ )(X◦ − Xl )◦ ∈ {d, q},′′′
where Xd , Xd′ , Xd′′ , Tdo, Tdo
denote d-axis synchronous reactance, transient reactance,′′′
subtransient reactance, and transient open-circuit, Xq , Xq′ , Xq′′ , Tqo, Tqo
denote similar
q-axis quantities, and Xl denote a stator leakage reactance (for the specific values,see [40]). It should be noted that the parameters of the synchronous machine areexactly the same for all sorts of generators. The matrices of the turbine with governorin (4.4) for thermal power plants and nuclear power plants are given as




−5 0
0
0
125
 −5 −5 0
 0 
0 






Aζ =  0,4 −4 0  , Bζ = 
 0 


1
1
0
−
0
0
9
9
Rζ = [0 5 0 0]T , Cζ = [0 0 0.3 0.7] .
The matrices for hydroelectric power plants is given by








0 
 −0.01 −0.25 0 −0.25 




2.3
 − 1



1
−
0
0 

 − 80 
2000
24
 , Bζ = ,Aζ = 




1
1



0
−
0 
0 



7
7




2
2
0
0
0
−
3
3

T
1
Rζ = 0 0 0 , Cζ = [0 0 − 2 3] .
7
The matrices for the excitation system with AVR in (4.6) are given as



−5
0
−5


Aη =  50 −0.5 0  ,10 −0.1 −2




5
 
Bη =  0  ,0Rη = [0 0.5 0.1]T ,Cη = [0 1 0] .

It should be noted that all sorts of synchronous generator have the same parameterswith respect to the excitation system as synchronous machine.

46

4.2.2

Load, PV Power Plant, and Non-Unit Bus

We adopt constant impedance load models [41]. For k ∈ NL , the model of the kth load
is given byi[k] = Y [k] v [k] , k ∈ NL
(4.7)
where Y [k] denotes the admittance of the load associated with the kth bus.
Next, the model of the PV generators is given. First of all, we assume that all PV
generators in every PV power plant are identical. Then the power generated by the
PV power plant is the sum of the output of the all PV generators. Every PV generator
is composed of a PV panel, a direct current (DC) /DC converter with a maximum
power point tracking (MPPT) controller, and an inverter with a controller [42]. Dueto the MPPT controller, the generated power from the converter is set to be constant.
Since the dynamics of the inverter is significantly fast due to the existence of thecontroller [43], it is reasonable to assume that the output of the PV plant is given by
constant power. Consequently, the model of the kth PV power plant is given byi∗[k] v [k] = pconst[k] ,k ∈ NP

where pconst
denotes a constant of the power output.
[k]Finally, from Kirchhoff’s low, the model of the kth non-unit bus is given byi[k] = 0,

4.2.3

k ∈ NN .

(4.8)

System Model

Now, suppose that the power grid is managed by multiple independent system operators(ISOs) and each component is associated with a single ISO. Let L denote the index setof ISOs and Il be the index of the components managed by the lth ISO. Then[

Il = N,l∈L

is satisfied, that is, there is no overlap among them.

47
(nl)

Let Σl denote the nonlinear model of the components associated with the lth(nl)ISO. The model Σl can be written by ẋl = fl (xl , v l , il , us,l , ul )
(nl)
Σl :
0 = g(xl , v l , il )
yl = hl (xl , v l , il )

(4.9)

where
xl := col(x[k] )k∈Il , v l := col(v [k] )k∈Il , il := col(i[k] )k∈Il ,us,l := col(us,[k] )k∈Il , ul := col(u[k] )k∈Il , yl := col(y[k] )k∈Il ,and fl and gl follow the dynamics of the components associated with the lth ISO.
From (4.2), il is given byXil :=Y lm v m(4.10)m∈Nlwhere Nl denotes the neighborhood of the components associated with the lth ISO andY lm denotes the corresponding admittance matrix. From (4.10), (4.9) can be written
by

!
X




ẋl = fl xl , v l ,Y lm v m , us,l , ul




m∈N
l


X

(nl)
0
=
g(x
,
v
,
Y lm v m )ll(4.11)
Σl :

m∈Nl



X


 yl = hl (xl , v l ,Y lm v m ).




m∈Nl
We here focus on small signal stability and it is suffices to consider the linear model

X

ẋ
=
A
x
+
L
Y lm v m + Bs,l us,l + Bl ul

l
l
l
l



m∈N

l


X

0 = Γl xl + Λl
Y lm v mΣl :(4.12)

m∈N
l


X




y
=
C
x
+
D
Y lm v m ,
l l
l

 l
m∈Nl

which is obtained from the linearization around an equilibrium. We consider the linear
model (4.12) hereinafter.
Our goal is to establish a distributed design method for PSSs that amend thetransient stability. The existing controller design approach [21] applies a traditional
controller design method such as optimal controller design for the associated subsystem

48
by treating the other part as a static system. As stated in Section 4.1, the stabilityof the whole power grid is not guaranteed when applying the existing approach. We
will see the case where the power grid loses the stability in Section 4.4 numerically. To
cope with the problem, we aim at build a systematic method for distributed design of
PSSs. The next section proposes such a distributed design method via retrofit control
developed in the previous chapters.

4.3

PSS Design based on Retrofit Control

This section provides a mathematical formulation of the PSS design problem statedin Section 4.2. Subsequently, we briefly review retrofit control in time-domain basedon state-space representation for developing a solution to the formulated problem. At
last, a PSS design approach based on time-domain retrofit control is proposed.

4.3.1

Problem Formulation

Let Xl denote the set of initial value of xl that can be possibly taken. Our aim is todesign a controller that provides ul so as to satisfy∥xL ∥2 ≤ γϵl ,∀(4.13)l∈L

for any xl (0) ∈ Xl andxl (0) = [0T · · · xl (0)T · · · 0T ]T
where(4.14)

xL := col(xl )l∈L ,ϵl > 0 denotes a desired performance, and γ > 0 is an independent constant of ul .
The measurement signal for the lth controller is restricted to only the local measurement output yl , the interconnection signal {y m }m∈Nl , and the supplementary control
input us,l alone. More specifically, the control input ul is given by
ul = Kl (yl , {v m }m∈Nl , us,l ),l∈L(4.15)

with a controller Kl . Moreover, we assume that all controllers are designed in a
distributed fashion. The controller Kl depends only on the elements of the set
Θl := {Al , Bs,l , Bl , Cl , Dl , Ll , Γl , Λl , {Y lm }m∈Nl } .

49Based on the conditions, the problem is formulated as follows.
Problem: Consider the power gird consisting of Σl for l ∈ L of which dynamics isrepresented as (4.12). It is assumed that the power grid is stable when ul = 0 for any
l ∈ L. Design Kl in (4.15) for l ∈ L so as to satisfy the following conditions:
1. The whole power grid becomes stable.
2. The performance criterion (4.13) holds.
3. Each control input can be represented by (4.15).
4. Each controller Kl is designed based only on their corresponding system parameters of Θl .

4.3.2

Hierarchical State-Space Expansion for Retrofit Control
in Time-Domain

We consider applying the retrofit control approach to the problem formulated above.
However, Assumption 1, which has been made throughout the previous chapters, does
not necessarily hold in power systems. To resolve this issue, hierarchical state-space
expansion for retrofit control in the time-domain is briefly reviewed in this subsection.
It is shown that retrofit control in time-domain does not require Assumption 1 but
provides us the same advantage as in the frequency-domain. We first show the importantlemma for the following discussion.
Lemma 1 For the systems(

andẋ = Ax + B0 u0 + B̂uy = Cx(4.16)˙ ξˆ = Âξˆ + B̂ ŵξ˙ = Aξ + (A − Â)ξˆ + B0 w
y = Cx(4.17)

where Â is any compatible matrix. Ifˆx(0) = ξ(0) + ξ(0)andu0 = w,

u = ŵ

50thenˆ
x(t) = ξ(t) + ξ(t),∀(4.18)t≥0

is satisfied for any w and ŵ.
Proof: The sum of the dynamics in (4.17) leads to
˙ + ξ(t)ˆ˙ = A ξ(t) + ξ(t)ˆξ(t)+ Bw + B̂ ŵ.
From the assumption, (4.18) is satisfied.

2

Lemma 1 implies that the state x can be obtained by summing of the states ξ andˆ
ξ. Due to Lemma 1, we can analyze the stability and L2 performance of x through theˆ We call the system representation (4.17) as hierarchical state-space
analysis on ξ and ξ.
expansion of the system (4.16).
It is assumed that a controller for w = u0 is attached in advance and its dynamics
is given byξ˙c = Ac ξc + Bc y




ˆ
= Ac ξc + Bc C(ξ + ξ)
(4.19)
w=
Cc ξc + Dc yˆ= Cc ξc + Dc C(ξ + ξ).
Then the dynamics of ξ and ξc is represented as
"

where

"
ACL :=

ξ˙
ξ˙c

#

"
= ACL

A + BDc C BCc
Bc C
Ac

ξ
ξc

#
+ R0 ξˆ

#

"
,

R0 :=

A − Â + BDc C
Bc C

#
.

The following lemma gives a controller design criterion for guaranteeing the stabilityof the system (4.17).
Lemma 2 Assume that ACL is stable. Consider the controller
(
K̂ :

˙
ξˆc = Âc ξˆc + B̂c Ĉ ξˆ
ŵ = Ĉc ξˆc + D̂c Ĉ ξˆ

(4.20)

51
where Ĉ := C. If the matrix"
ÂCL :=Â + B̂ D̂c Ĉ B̂ Ĉc
B̂c Ĉ
Âc#

is a stable matrix, then the entire system (4.17) with the controllers (4.19) and (4.20)
becomes stable.
Proof: The whole system is represented as˙"#   ξˆ 
ξˆ
0
0
 ˙ 
ÂCL
 ξˆc  

ξˆc 


=

0
0
.

 ˙   hi ξ 
 ξ ACL
R0 0 0ξcξ˙c

Because of the block triangular property of the matrix,



σ
 h

"
ÂCL
R0 0 0

i

0 0
0 0

# 

ACL




 = σ ÂCL ∪ σ (ACL ) .
The assumption leads to the stability of the entire system.

2

Lemma 2 implies that the controller (4.20) stabilizes the system (4.17) with thecontroller (4.19) as long as the upstream dynamics of (4.17) with the controller (4.20)
is stable. Thanks to the cascade structure, the stability is assured with the controllers
ˆ As a result, the original system becomes stable
for the upstream part with respect to ξ.
by employing the control input that stabilizes the upstream parts provided by K̂ from
Lemma 1. A strategy to generate the control input is included in the discussion in thenext subsection.
Finally, the following lemma is shown in terms of performance of the entire system.

Lemma 3 Assume that ÂCL is a stable matrix and the initial values hold
ˆ = x(0),ξ(0)ξ(0) = 0.

52
Suppose the controller (4.20) is implemented. Then the bound of L2 -norm of x is givenby∥x∥2 ≤ γ0 ϵ̂
where
γ0 := ∥ [I 0] (sI − ACL )−1 R0 + I∥∞ ,ˆϵ̂ := eÂCL t ξ(0)2ˆˆ in the
Proof: Let x(s), ξ(s) and ξ(s)denote the functions of x(t), ξ(t), and ξ(t)Laplace domain. Lemma 1 leads to the inequality∥x(t)∥2 = ∥x(s)∥2ˆ= ∥ξ(s) + ξ(s)∥2ˆ + ξ(s)∥ˆ= ∥ [I 0] (sI − ACL )−1 R0 ξ(s)2ˆ≤ γ0 ∥ξ(s)∥2= γ0 ϵ̂and hence the conclusion holds.

2

It should be noted that γ0 is an independent constant of the controller and ϵ̂ isdependent only on ÂCL . This result implies that the size of the state x is possibly
suppressed in the sense of the upper bound by designing a controller that effectively
works for the upstream part in (4.17). We utilize these results to develop retrofit
control in time-domain. In the next subsection, we propose a PSS design method via
retrofit control in time-domain that the results seen in this subsection produce.

4.3.3

Proposed PSS Design Method

First of all, let us try to determine K̂ in (4.20). From (4.12), we havev L := ΠL xL
where xL is the states of the entire system given as (4.14), andv L := col(v l )l∈L , ΠL := −(ΛY L )−1 ΓL ,
ΛL := diag(Λl )l∈L , ΓL := diag(Γl )l∈L .

(4.21)

53
Substituting (4.21) to (4.12), we have the model of the whole power gridẋL = AL xL + Bs,L us,L + BL uL y =C x
L
L L
ΣL :

ẋs = Cs xLus,L = κs axs
whereuL := col(ul )l∈L , yL := col(yl )l∈L , AL := diag(Al )l∈L + LL Y L ΠL ,BL := diag(Bl )l∈L , Bs,L := diag(Bs,l )l∈L , LL := diag(Ll )l∈L ,CL := diag(Cl )l∈L + DL Y L ΠL , DL := diag(Dl )l∈L ,(4.22)

xs denotes the state of the supplementary controller, Cs denotes the matrix associatedwith the aggregated frequency deviation, a := col(a[k] )k∈NG , and us,L := col(us,l )l∈L .
Now, set"#Ac,l Bc,lK̂l :=
Cc,l Dc,land choose matrices Ac,l Bc,l , Cc,l , Dc,l for l ∈ L according to the following criterion
eÂCL,l t xl (0)where"ÂCL,l :=2≤ ϵl ,∀(4.23)

xl (0) ∈ Xl

Al + Bl Dc,l Cl Bl Cc,l
Bc,l ClAc,l#.

The controller design problem is a standard H2 suboptimal control problem that canbe readily solved using some techniques, e.g., linear matrix inequalities, (for example,see [44]). The following theorem holds.
Theorem 8 Consider the system (4.12). Design K̂l according to the performance
criterion (4.23). Set ÂL , B̂L , ĈL , and γ byÂL := diag(Al )l∈L ,B̂L := diag(Bl )l∈L ,ĈL := diag(Cl )l∈L ,andγ := [I 0](sI − AL,CL )−1 R + I ∞

54
where

"
AL,CL :=

AL Bs,L κs a
Cs
0

#

"
,

R :=

AL − ÂL
Cs#.

Further, let the controller (4.20) as
Ac := diag(Ac,l )l∈L ,Cc := diag(Cc,l )l∈L ,Bc := diag(Bc,l )l∈L ,Dc := diag(Dc,l )l∈L .

Then the L2 -norm of the state can be bounded by (4.13).
ˆ = x(0),Proof: Because ξ(0)ˆeÂcl t ξ(0)=2X

eÂcl,l t xl (0)

l∈L

.
2

Thus, for arbitrary x0 (0) ∈ Xl ,ˆeÂcl t ξ(0)2= eÂcl,l t xl (0)2is satisfied whenx(0) = [0T xl (0)T 0T ]T .
Lemma 3 indicates thatˆ
∥xL ∥2 ≤ γ eÂCL t ξ(0)2= γ eÂCL,l t xl (0).
2The assumption leads to that∥x∥2 ≤ γϵlfor any x0 (0) ∈ Xl .

2

Theorem 8 provides a design strategy for Ac,l Bc,l , Cc,l , Dc,l so as to give a performancebound according to (4.13). Pursuing the procedure developed above, we can set K̂lunder the constraint on distributed design.
Next, we discuss implementation of the designed controllers. Since the measurement
signal of the controllers is Ĉ ξˆ as represented by (4.20) and ξˆ becomes a virtual signal,
the measurement signal cannot be obtained. To resolve the difficulty, under the variable55
transformation

"

xL
x̂L

#

"
=

I I
0 I

#"

ξˆ
ξ

#
(4.24)

we consider the systemẋL = AL xL + Bs,L us,L + BL uL ẋ = C xss Lus,L = κs axs ˙x̂L = ÂL x̂L + (AL − ÂL )xL + Bs,L us,L .

(4.25)The inverse of the coordinate transformation (4.24) leads to thatξˆ = xL − x̂L .
Therefore if x̂L in (4.25) is available in a distributed fashion then we can obtain thecontrol input in (4.20) employing x̂L . We refer to the system providing x̂ as an output
rectifier and discuss a design method for decentralized output rectifier.
Note thatAL − ÂL = LL Y L ΠLand since AL − ÂL does not hold block diagonal structure and (AL − ÂL )xL cannot
be created using xL in a distributed fashion, it is impossible to design and implementa decentralized output rectifier providing x̂L using xL . To overcome the problem, we
employ not the signal xL but the interconnection signal v L . The relation(AL − ÂL )xL = LL Y L ΠL xL= LL Y L v LP= col Ll m∈Nl Y lm v m l∈L ,(4.26)

leads us to following theorem.
Theorem 9 For the systems (4.25) and
ẋL = AL xL + Bs,L us,L + BL uL ẋ = C xss Lus,L = κs axs


P
 ˙
x̂L = ÂL x̂L + Ll m∈Nl Y lm v m + Bs,L us,L ,(4.27)l ∈ L,

56
set the initial valuesx̂L (0) = 0
andl ∈ L.

x̂l (0) = 0,
Thenx̂L (t) = col(x̂l (t))l∈L ,∀t≥0

is satisfied for any control input uL .
Proof: Let ê the error and its dynamics is given byê := x̂L − col(x̂l )l∈L .
From (4.21), (4.22), and (4.26), the dynamics is simplified asê˙ = ÂL ê.
The assumption on the initial condition leads to thatê(t) = 0,∀

t ≥ 0.

Thus, x̂L (t) = col(x̂l (t))l∈L for any t ≥ 0.

2

Theorem 9 indicates that the decentralized output rectifiers(
GR,l :Px̂˙ l = Al x̂l + Ll m∈Nl Y lm v m + Bs,l us,l,Pŷl = yl − Cl x̂l + D m∈Nl Y lm v ml∈L

can generate the measurement outputĈ ξˆ = Ĉ (xL − col(x̂l )l∈L )= ŷLwhere ŷL := col(ŷl )l∈L and enable us to create the control input in (4.20).
Finally, under the initial conditions
ˆx(0) = ξ(0),ξ(0) = 0,x̂l (0) = 0,l ∈ L,(4.28)

57

Fig. 4.2 Block diagram of the local system structure composed of Σl , GR,l , and K̂l .
letting A = AL , B = Bs,L , B̂ = BL , and C = CL we can build an equivalent system
consisting of Σl , GR,l , K̂l for l ∈ L to the whole system with (4.17) with (4.20), whereΣl is given by (4.12), GR,l is given by (4.28), and(K̂l :ẋc,l = Ac,l xc,l + Bc,l ŷl,uc,l = Cc,l xc,l + Dc,l ŷll ∈ L.

(4.29)

It is shown that K̂l can be designed only with the parameters Al , Bl , Cl , and Dlfrom (4.23) and GR,l and K̂l can be designed and implemented in a distributed fashion
from (4.28) and (4.29). Fig. 4.2 depicts the block diagram corresponding to Σl , GR,l ,and K̂l .
We here summarize the obtained results and the proposed approach. The design method is given as follows. Choose Ac,l Bc,l , Cc,l , Dc,l such that the performance
criterion (4.23) is satisfied. Establish the controller Kl (yl , {v m }m∈Nl , us,l ) as
(Px̂˙ l = Al x̂l + Ll m∈Nl Y lm v m + Bs,l us,l



P

 GR,l :ŷl = yl − Cl x̂l + Dl m∈Nl Y lm v m(Kl :ẋc,l = Ac,l xc,l + Bc,l ŷl K̂l : u = C x + D ŷ .
c,lc,l c,lc,l l(4.30)

58
We can confirm that the conditions of the formulated problem are satisfied as follows.
1. Because the grid with the supplementary controller becomes stable, Lemma 2
implies that the entire system with the designed PSSs is stable.
2. Since the obtained whole system is equivalent to the system (4.12) with (4.20),
Theorem 8 indicates that the performance criterion (4.13) is satisfied.
3. From the form of the design controller (4.30), it can be seen that the measurement
signals are yl , {v m }m∈Nl , and us,l .
4. From the design process of the controllers, only the local parameters in the set
Θl are used and hence distributed design is achieved.
Finally, it should be remarked that the form of the retrofit controller (4.30) isexactly the same as that of retrofit controllers developed in the previous chapters whenregarding Σl as the local system and (v m , us,l ) as the interconnection signal from theenvironment. However, for the development of the controller (4.30), a retrofit controller
in the time-domain, the assumption that the local system is stable is not made atall. Thus we can apply the retrofit control approach for power systems in which local
systems are not necessarily stable.

4.4

Numerical Simulation

This section provides a numerical simulation showing the effectiveness of the proposed
approach for control of power grids. PV-integrated EAST30 model is used and thegird is depicted in Fig. 4.3. The PV-integrated EAST30 model is composed of 137
buses, 30 generators, 31 loads, 30 PVs, and transmission facilities. The transmission
line parameters of EAST30 model are given in [25]. Although no PV generators are
installed into the original IEEJ EAST30 model, in order to simulate future powersystems we place some PV power plants. The places of the buses where PV power
plants are attached are determined according to the data of areas where large amountof PV is penetrated in the future Japanese power system [45]. Since the model of thegenerator has 13-dimensional states, the dimension of the whole grid reaches at 390.
The PV generation is assumed to cover 6% of the total demand in the grid. The amountis supposed to be 30% of the FIT installed amount in Japan. The supplementary
control is designed based on the following parametersκs = −1,a[k] = 1.

∀k ∈ NG .

59

(25)

G25

G20

(72)

G22

G21

(20)

(21)

(22)

(66)

(67)

(68)

(73)

(8)

(53)

(26)

G26

(65)

G9

(54)

(69)
(107)

(51)

(100)

(24)

(7)

(38)

(49)

G10

(112)

(76)

G27

(52)

(55)

(75)

(70)
(23)

(97)

G11

(6) (48)

(84)

(5)

(98)

(102)

(132)

G29

(86)

(85)

(120)

(95)

(13)

(42)
(114)

G4

(130)
(134)

(118)

(60)
(127)

(56)

(59)

(88)

(96)

(16)

(30)

(19)

G19

(80)

(89)

(90)

(17)

(18)
(123)

(32)

(58)

(91)

(135)
(136)

(40)

G14

(61)

(106)

(46)

(122)

(81)

(105)

(33)

(41)

(14)

G16
G30

(110)

(4)

(104)

(133)

(34)

(115)

G13

(99)

(79)

(103)

(111)

(119)

(87)

(129)
(62)

(29)

(28)

(47)

G5

(121)

(78)

(108)

(35)

G12

(63)

(71)

G28

G2

(12)

(93)

(94)

(2)

(44)
(43)

(82)

(83)

(77)

(126)

(37)

G6

(128)

G23

(131)

(36)

(116)

(11)

(64)

(101)

(3)

G3

(113)

G7

(10)

G24
(27)

(117)

(9)

(137)
(74)

(39)

(45)

G8

(50)

G18

G17

(124)

(57)

(31)

(15)

(92)

G15

(109)

G1

(1)

(125)

Fig. 4.3 PV-integrated IEEJ EAST 30-machine power system.
The whole grid is stable with the supplementary controller unless any additional
controller is implemented.
The indices of the components are given by
L = {1, 2},
I1 ∩ NG = {1, . . . , 5},
I1 ∩ NL = {37, 40, 41, 42, 44},
I1 ∩ NP = {112, 117, 122, 127, 132},
I1 ∩ NN = {31, . . . , 36, 38, 39, 43, 50, . . . , 58, 60, . . . , 68},
I2 = {1, . . . , N } \ I1 .
Let us suppose a fault happening in the second generator. The model of the contingency
is given by an initial disturbance injected into the angle of the corresponding generator,
namely,(0.1e1 if k = 2,x[k] (0) =0otherwise.

60
For the given power grid, consider attaching retrofit controllers developed in theprevious sections. We design Ki in (4.30) for I1 utilizing the H2 optimal control where
the objective function is given by
Z ∞
J1 :=


xT1 Q1 x1 + ruT1 u1 dt

0

where
Q1 = qδ,1 ê1 êT1 + qω,1 ê2 êT2 + q1 I
andêi := 1|I1 ∩NG | ⊗ ei ,1M is the M -dimensional all-ones vector, ei ∈ R13 for i ∈ {1, 2} is the canonical bases
associated with the ith coordination, and qδ,1 , qω,1 , q1 and r are positive constants. It isnotable that qδ,1 and qω,1 are associated with the angle and the frequency, respectively.
For the second region I2 , we place no primary controller to the power grid. We
have employed Matpower [46] on MATLAB R2017a for the following numericalsimulations.
We consider the three cases: First, no controller is added except for the supplementary controller. Second, a PSS designed with the traditional approach, namely,
single-machine infinite-bus system. Finally, a PSS designed with the proposed retrofit
control approach. The free responses of all generators’ frequency deviation for t ∈ [0, 50]
are shown in Fig. 4.4. It is shown that large oscillations happen when no control isadded. The responses for t ∈ [0, 1.5] under the traditional approach are shown in
Fig. 4.5. We can see that the power grid losses its stability by attaching the controller.
The responses under the proposed approach for t ∈ [0.50] are shown in Fig. 4.6 whereq1 = 1,qδ,1 = 103 ,qω,1 = qω,0 ,r = 1.

with qω,0 := 107 . The figure demonstrates that the system is kept to be stable evenwhen attaching the primary controller. Furthermore, the responses for t ∈ [0, 0.5] isshown in Fig. 4.7. The figure indicates that the peak of the frequency deviation issuppressed by the proposed approach from −0.0904 to −0.0753. In order to further
reduce the peak of the frequency deviation, we next design a controller using theparameterqω,1 = qω,0 × 103 .

61

Fig. 4.4 Free responses of all generators’ frequencies.
The responses with the newly designed controller are shown in Fig. 4.8. The figure
indicates that the oscillation is still more reduced to −0.0505, which is 55% of thepeak value without any controller. These results suggest that the proposed PSS design
method is effective for improving damping performance of the power grid.

4.5

Chapter Summary

In this chapter, we have proposed a distributed design method for PSSs to cope with
contingencies happening in power systems where a large amount of renewable energy
resources is installed based on retrofit control. Since power grids into which large PV
is introduced are susceptible to disturbance and readily destabilized, it is required todevelop a novel method for designing PSSs that guarantee the stability of the whole
power grid. While existing practical methods, including the single-machine infinite-bus
based method, cannot theoretically guarantee the stability, retrofit control can givetheoretical assurance of the stability. The proposed method is an extension of theexisting retrofit control to be applicable for power systems. We have first provided
a dynamical model of the power grid that consists of synchronous generators, loads,
PV plants, and buses. It has been shown that the power system can be treated as alarge-scale network system and the problem on distributed design of PSSs have been

62

Fig. 4.5 Responses of all generators’ frequencies with the controller designed based on
the single-machine infinite-bus system.
mathematically formulated. We have derived a solution to the formulated problem
based on retrofit control in the time-domain. Numerical simulations with PV-integrated
EAST30 model have shown that the proposed approach not only stabilizes the power
grid, which can be destabilized by a existing method using the single-machine infinitebus model, but also sufficiently suppresses the frequency deviation.

63

Fig. 4.6 Responses of all generators’ frequencies of the closed-loop system with theproposed method when qω,1 = qω,0 .

Fig. 4.7 Comparison of the responses without control and with the proposed method
when qω,1 = qω,0 .

64Fig. 4.8 Responses of all generators’ frequencies of the closed-loop system with theproposed method when qω,1 = qω,0 × 103 .

Fig. 4.9 Comparison of the responses without control and with the proposed method
when qω,1 = qω,0 × 103 .

Chapter 5
Glocal Control of Network Systems
via Hierarchical Representation
5.1

Introduction

We have developed a distributed design method for decentralized controllers. Basedon the results seen in the previous chapters, we generalize the approach to glocal(global/local) control, where spatially distributed decentralized controllers and a central
broadcasting controller cooperatively control the entire network system. We, in practice,take an objective-based approach to system modeling and control, e.g., subsystem
identification for local decentralized control and modeling of aggregate system behavior
for global broadcast control. We investigate such different kinds of control to realize
desired system behavior.
The concept of glocal control has been already introduced in [26] but a systematic
design of glocal controllers has not been established yet. We propose a novel glocal
control method based on hierarchical representation where the original system isexpressed as a hierarchical system. The approach for glocal control design is to consider
an equivalent hierarchical system in which the upstream and downstream parts represent
local and global behaviors, respectively. Then the original system is represented asa cascade system that consists of low dimensional systems. For the resulting system,
we design local and global controllers for the upstream and downstream parts. The
design procedure of controllers are independently performed and it indicates that thecontroller can be designed in a distributed manner.
Composite control, which is an existing approach employing multiple controllers
for systems having multi time-scale, e.g., power plants [47–50], network systems
composed of multiple robots [51], and transmission control protocol [52, 53]. The

66
composite control requires the premise on which the whole system dynamics canbe decomposed into two time-scales, slow and fast ones, by using the techniquecalled singular perturbation [54, 55]. Under singular perturbation, each dynamics istreated as global and local dynamics that represent corresponding aggregated andlocal behaviors, respectively [56]. Nevertheless, it is not theoretically shown that the
time-scale separation works well. The whole system, indeed, possibly loses the stability
if the gap of the time-scales is insufficient, due to unanticipated interference among
the controllers for the separated dynamics.
This chapter is organized as follows. We first see an example that motivates us
to develop glocal control in Section 5.2. In Section 5.3, we give the system model
treated in this chapter and the definition of hierarchical representation that is the
key to developing glocal control. Further, we derive a characterization of hierarchicalrepresentation for a given system from the viewpoint of controllable subspace. In
Section 5.4, we propose a distributed design method for designing glocal controllers
based on hierarchical representation. Moreover, in Section 5.5, we see the effectivenessof the proposed glocal control through a numerical example of power system control.
Finally, Section 5.6 concludes this chapter.

5.2

Motivating Example

First of all, a motivating example is shown to explain the framework of the proposed
glocal control.
Let us consider a network composed of three second-order systems. Each subsystem
is represented bymi θ̈i + di θ̇i + vi + ui = 0, i = 1, 2, 3
where vi is the interconnected signal injected into the ith subsystem and ui is thecontrol input. The interconnection signal is represented byvi =Xkij (θi − θj ).

j̸=i

We assume that the system is homogeneous, that is, the coefficients are assumed to be
the same as the other subsystems’ ones and subscripts for all coefficients are omitted
in the following. Then the entire system can be represented by
"
ẋ =

I⊗

0
1
0 −d/m

#

"
−L⊗

0 0
1 0

#!

"
x+I ⊗

0
1

#
u

67
where
x :=

3
X

"

θi
θ̇i

ei ⊗

i=1

#
,

u :=

3
X

ei ui

i=1

with the canonical bases ei ∈ R3 and−kij /m, i ̸= j, X(L)ij :=
kij /m, i = j.
j̸=i

We suppose that the control input is represented as the sum of local control inputs
and a global control input byu=3
Xei uL,i + 13 uG

i=1

with all-ones vector 13 ∈ R3 .
We now consider the hierarchical system

"
#
" #

0
1
0


ξ˙i =
ξi +
uL,i , i = 1, 2, 3
−3k/m −d/m
1
"
#
"
#
" #
3

X

0
1
0
0
0

˙

ξi +
uG .

 ξ0 = 0 −d/m +−k/m 01
i=1Then it can be shown that
3X
x(t) = (13 ⊗ I)ξ0 (t) +(ei ⊗ I)ξi (t),

∀

t≥0

i=1

for all uL,i , uG provided thatx(0) = (13 ⊗ I)ξ0 (0) +3X(ei ⊗ I)ξi (0)i=1

by simple algebra.
We consider designing local and global controllers
uL,i = KL (ξi ),i = 1, 2, 3

68
anduG = KG (ξ0 ).
If the controllers stabilizes the corresponding system, the entire system is also stable
due to the cascade structure. As a result, the original system is also stabilized. Then
distributed design of glocal controllers is achieved.
Now a couple of questions are posed. The first question is what the condition
is for existing such a hierarchical system of which the superposition of the states
represents the original state exists. The second question is what the condition is forthe hierarchical system to model after the original system. In the next section, we
provide answers to the questions from the perspective of controllability.

5.3

Hierarchical Representation

In this section, we introduce the notion of hierarchical representation that is the key
to developing distributed design of glocal controllers. First of all, we give a precise
definition of hierarchical representation. Subsequently, we provide a characterization ofhierarchical representation to establish a hierarchical representation for a given system.

5.3.1

Definition of Hierarchical Representation

We consider linear time-invariant systems represented byΣ : ẋ = Ax +N
X

Bi ui + B0 u0

(5.1)

i=1

where x is the state and ui for i = 1, . . . , N are local control inputs and u0 is global
control input. Our aim here is to develop a distributed design method for local andglobal controllers for the control inputs.
Now we consider the following hierarchical system˙ ξi = Âi ξi + B̂i ui , i = 1, . . . , N
N
X
Ξ:
˙

R̂i ξi + B̂0 u0 .
 ξ0 = Â0 ξ0 +i=1

The definition of hierarchical representation is given as follows.

(5.2)

69
Definition 3 The hierarchical system (5.2) is called a hierarchical representation of
the original system (5.1), whenx(t) = P0 ξ0 (t) +NX

Pi ξi (t),

∀

t≥0

(5.3)

i=1

holds for any u0 , u1 , . . . , uN with full-column rank matrices P0 , P1 , . . . , PN whose sizeare compatible with the dimension of the states under the initial condition
x(0) = P0 ξ0 (0) +N
X

Pi ξi (0).

i=1According to the definition, the original state x can be represented as a superpositionof the states ξi provided that the system (5.2) is a hierarchical representation of (5.1).
Once a hierarchical representation is given, local controllers generating ui for i =1, . . . , N can be designed only with the parameters Âi and B̂i and a global controller
can be designed only with Â0 , R̂i , B̂0 as well. In this sense, hierarchical representation
greatly helps us to achieve distributed design of glocal controllers.

5.3.2

Characterization of Hierarchical Representation

We give a characterization of hierarchical representation in this subsection.
First, we give a condition for existence of hierarchical representation. The following
theorem holds.
Theorem 10 There exists a hierarchical representation of the system Σ in (5.1) if
and only if R(A, Pi ) ⊂ im Pi + im P0 , i = 1, . . . , N(5.4)
R(A, P0 ) ⊂ im P0 ,

im Bi ⊂ im Pi , i = 0, . . . , N
holds.
Proof: Sufficiency is shown in the proof of Theorem 11 by constructing a specifichierarchical representation. we here show necessity in this proof.
Assume that Ξ is a hierarchical representation of Σ. Let
x0 ∈ im P0

70
andx(0) = x0 ,

ξ0 (0) = P0† x0 ,ξi (0) = 0

for i = 1, . . . , N . Because Ξ is a hierarchical representation,x(t) = P0 ξ0 (t),∀t≥0holds for ui = 0 for i = 0, . . . , N . Thus∀x(t) = eAt x0 ∈ im P0 ,t ≥ 0.

Since im P0 is a closed subspace,eAt x0 − x0
lim
= Ax0 ∈ im P0 .
t→0t
Because x0 is an arbitrary element in im P0 , im P0 is an invariant subspace of A. Hence
R(A, P0 ) ⊂ im P0 .
Similarly, it can be shown that R(A, Pi ) ⊂ im P0 + im Pi for i = 1, . . . , N by taking
x0 ∈ im Pi .
Finally, we show that im Bi ⊂ im Pi for i = 0, . . . , N . Fix i ∈ {1, . . . , N } and set
x(0) = 0, ξi (0) = 0 for i = 0, . . . , N, and uj = 0 for j ̸= i. Thenẋ(0) = Bi ui (0) ∈ im Bi
for any ui . On the other hand,ẋ(0) = P0 ξ˙0 (0) +NX

Pi ξ˙i (0)

i=1

= P0

Â0 ξ0 (0) +

N
X
i=1

!
R̂i ξi (0) + B̂0 u0 (0)

+

N
X



Pi Âi ξi (0) + B̂i ui (0)i=1= Pi B̂i ui (0) ∈ im Pi .
Because there exists ui for any ẋ(0) ∈ im Bi , im Bi ⊂ im Pi , for i = 0, . . . , N .

2A signal-flow diagram of the hierarchical system Ξ is shown in Fig. 5.1 where Ξicorresponds to the dynamics of ξi for i = 0, . . . , N . As shown in the figure, ξi for
i = 1, . . . , N does not affect ξj for j ̸= i at all and the necessity indicates that theoriginal system must hold the structure.

71

Fig. 5.1 Signal-flow diagram of Ξ.
We next give a characterization for Ξ to be a hierarchical representation of Σ. The
following theorem holds.
Theorem 11 The system Ξ is a hierarchical representation of Σ if and only if† Â0 = P0 AP0
APi − P0 R̂i − Pi Âi = 0, i = 1, . . . , N
Pi B̂i = Bi , i = 0, . . . , N(5.5)

holds.
Proof: (Sufficiency) First of all, we show that there exist matrices Âi and R̂i for
i = 1, . . . , N such that the condition (5.5) holds when the condition (5.4) is satisfied.
Since
R(A, Pi ) ⊂ im Pi + im P0 , i = 1, . . . , N
holds,
im APi ⊂ im Pi + im P0 ,i = 1, . . . , N.

Hence there exist Xi , X0 such thatAPi = Pi Xi + P0 X0 .

72
Thus
Âi = Xi ,R̂i = X0

satisfy the condition (5.5).
Let us assume that the condition (5.5) holds. Define
e := x − P0 ξ0 −

N
X

Pi ξi

i=1

and thenė = Ax − P0 Â0 ξ0 += Ax − P0 Â0 ξ0 −NX!
R̂i ξi−NX

Pi Âi ξi

i=1

i=1

N 
X



P0 R̂i + Pi Âi ξi

i=1= Ax − P0 P0† AP0 ξ0 −N
XAPi ξi .

i=1

Since R(A, P0 ) ⊂ im P0 ,P0 P0† AP0 = AP0 .
Thereforeė = Ax − AP0 ξ0 −N
XAPi ξi
i=1!
NX= A x − P0 ξ0 −
Pi ξi
i=1

= Ae.
When x(0) = P0 ξ0 (0) +i=1 Pi ξi (0), e(0) = 0 and hencePNe(t) = 0,∀t≥0

for any ui for i = 0, . . . , N . Since there is no error, (5.3) holds. Thus Ξ is a hierarchicalrepresentation.
(Necessity) First we show that Â0 = P0† AP0 , P0 B̂0 = B0 . When ui = 0 and ξi = 0
for i = 1, . . . , N , ξi (t) = 0 for any t ≥ 0. Thus thenx(t) = P0 ξ0 (t),

∀

t≥0

73
for any u0 provided that x(0) = P0 ξ0 (0). Define
e0 := P0† x − ξ0and thenė0 = P0† Ax + P0† B0 u0 − Â0 ξ0 − B̂0 u0= P0† AP0 P0† x − Â0 ξ0 + (P0† B0 − B̂0 )u0= P0† AP0 e0 + (P0† AP0 − Â0 )ξ0 + (P0† B0 − B̂0 )u0 .

Since e0 (t) = 0 for ant t ≥ 0, ė0 (0) = 0 for any t ≥ 0. Becauseė0 (0) = (P0† AP0 − Â0 )ξ0 (0) + (P0† B0 − B̂0 )u0 (0)for any ξ0 (0) and u0 (0), Â0 = P0† AP0 , P0 B̂0 = B0 holds. Similarly, it can be shownthat APi − P0 R̂i − Pi Âi = 0 and Pi B̂i = Bi , 1 = 0, . . . , N .
2
Theorem 11 provides a characterization of hierarchical representations from theviewpoint of controllability. Some specific examples are given here.
Example 1: Let us suppose that
im Pi ⊂ im P0 ,i = 1, . . . , N.

Then there exists R̂i such that the condition in (5.5) is satisfied for any Âi . Thus all
hierarchical representations are parameterized by†˙ ξi = Âi ξi + Pi Bi ui ,i = 1, . . . , N
N 

X
Ξ:
†
†
†
˙

ξ
=
P
AP
ξ
+
P
AP
−
P
Â
0 0
i
i i ξi + P0 B0 u0
 0
0
0
i=1

with Âi for i = 1, . . . , N . Note that this is exactly the same as the retrofit controller inthe time-domain employed in Chapter 4 when P0 = I.
Example 2: Let us suppose that
im Pi ∩ im P0 = {0},i = 1, . . . , N.

74
Then Âi and R̂i are uniquely determined. Accordingly,†˙ ξi = Xi ξi + Pi Bi ui , i = 1, . . . , N
N
X
Ξ:
†
˙

X0i ξi + P0† B0 u0 ξ0 = P0 AP0 ξ0 +i=1

is the unique hierarchical representation where Xi and X0i are the matrices††

X0 = (P 0 Pi )† P 0 APi ,††X0i = (P i P0 )† P i APi ,which satisfy
APi = P0 X0i + Pi Xi .
Through the following sections, we develop a glocal control method via hierarchicalrepresentation.

5.4

Glocal Control for Network Systems

In this section, we provide a distributed design procedure of glocal controllers via
hierarchical representation.

5.4.1

Difficulty on Glocal Controllers Implementation

To clearly express the network structure, we consider a linear network system composedof N subsystems ẋi = Ai xi + Li vi + Bi ui
Gi :w i = Γ i xi
y i = C i xi
with the network
w = Mvwhere
w := col(wi ),v := col(vi ).

75
We suppose that the control input can be decomposed by
u := col(ui )NX=(ei ⊗ I)uL,i + 1N uG
i=1with local control inputs uL,i and a global control input uG . Then the entire network
system is written by(ẋ = Ax + Bu
Gall :(5.6)
y = Cxwhere
A := diag(Ai ) + diag(Li )M diag(Γi ),B := diag(Bi ),C := diag(Ci ).

We now assume that Ξ in (5.2) is a hierarchical representation of Gall with matrices
P0 , P1 , . . . , PN , wherePi := ei ⊗ I.
It is easy to design controllers Ki for i = 1, . . . , N only with the model of Gi such thatthe system(ξ˙i = Âi ξi + B̂i uL,i
(5.7)
uL,i = Ki (Ĉi ξi )is stable where Ĉi := Ci . It is also easy to design a controller K0 such that the system(ξ˙0 = Â0 ξ0 + B̂0 uG
uG = K0 (Ĉ0 ξi )is stable whereĈ0 :=

NX(5.8)(eTi ⊗ I)Ci .

i=1

Owing to the cascade structure of a hierarchical representation, stability of the entire
system can be guaranteed if the control inputs in (5.7) and (5.8) can be implemented.
The difficulty here is that the measurement signals Ĉi ξi cannot be directly measured.
We will solve the problem in the next subsection.

76

5.4.2

Rectifier Design

The main idea to avoid the problem above is to design a linear function observer that
produces the signals Ĉi ξi , which we call a rectifier as an analogy of the rectifier inretrofit controller design.
The following theorem holds.
Theorem 12 Consider the system (5.6) and assume that Ξ in (5.2) is a hierarchicalrepresentation of Gall . Assume also that Ki for i = 0, . . . , N are designed so as tostabilize the systems (5.7) and (5.8). Then the system Gall with the controllers
(

and

(
Gi :

uL,i = Ki (Ĉi ξi )uG = K0 (y)ϕ̇i = Ai ϕi + Byi yi + Bvi vi + Bui ui + Bu0 u0ξˆi = Ci ϕi + Di yi

for i = 1, . . . , N is stable if Gi is a linear function observer of Ξ in (5.2) for ξi .
Proof: We assume that Gi is a linear function observer for ξi . From the necessary
and sufficient condition [57], it suggests that Ai is stable for any i and there exists a
matrix U such that "#diag Âi
0U− diag (Ai ) U = diag (Byi Ci + Bvi Γi ) [I P0 ] ,R̂1 · · · R̂N Â0
 
"
#
(5.9)
diag B̂i0[diag (Bui ) 1N ⊗ B0 ] = U,0B̂0diag (Di ) [diag(Ci ) P0 ] + diag(Ci )U = [I 0] .
From the second equation in (5.9), it turns out that U has the form
U = [diag (Ui ) 1N ⊗ U0 ]as long as B̂i is a column-full rank matrix for any i.
Defineei := Ui ξi + U0 ξ0 − ϕi .

77Then because Gi is a linear function observer, the dynamics can be representedėi = Ai ei
which is stable. Now sinceξˆi = ξi − Ci ei
from the conditions in (5.9), the entire closed-loop system is composed of ėi = Ai eiξ˙i = Âi ξi + B̂i ui
ui = Ki (Ĉi (ξi − Ci ei ))for i = 1, . . . , N , and(Pξ˙0 = Â0 ξ0 +  N
i=1 R̂i ξi + B̂0 u0 
PN.
uG = K0 Ĉ0i=1 Pi ξi + P0 ξ0

From the assumption and the hierarchical structure, the entire system is stable.

2

Because the controllers can be designed based only on the local models, if we candesign linear function observers only with the local model then distributed design ofglocal controllers is achieved. In fact, the following theorem holds and states that there
exist such rectifiers.
Theorem 13 Assume that Ai and Âi are stable. Let† ϕ̇i = Âi ϕi + (Ai − Âi )x̂i + Li vi + Pi P0 B̂0 u0Gi :x̂˙ i = Ai x̂i + B̂i ui + Li vi + Pi† P0 B̂0 u0 ˆ
ξi = −Ĉi ϕi + yi .
Then Gi is a linear function observer for Ci Pi ξi .
Proof: It suffices to show that there exists a matrix U such that"U diag Âi

0

#

"

 

 #
diag Âi diag Ai − Âi−R̂1 · · · R̂N Â0 0diag(Ai ) #"#"diag Ĉidiag Ĉi P0
0 diag (Li )=0 diag (Li )M diag (Γi ) M diag (Γi ) P0 ,

U

78
"

0

 P0 B̂0diag B̂i P0 B̂0

#

"
=U

 
diag B̂i

0

0

B̂0

#
,

and
"
[I 0]

 
diag Ĉi #diag Ĉi P0

M diag (Γi ) M diag (Γi ) P0
Let us take

"
U=

h
 i
h
  i
+ −diag Ĉi U = diag Ĉi 0 .

0 P0
I P0

#

and then simple algebra leads to the conclusion.

2

All parameters in Gi are associated with Gi and hence it can be designed in a
distributed manner.
The procedure of distributed design of glocal controllers via hierarchical representation is summarized as follows. First, we search for P0 , . . . , PN such that theconditions (5.4) hold for the system Gall . Next, parameters for a hierarchical representation are determined according to the conditions (5.4). For the given hierarchicalrepresentation, design controllers in (5.7) and (5.8). Finally, the resulting glocal
controllers are implemented with the rectifiers.

5.5

Application to Power System Control

This section gives a numerical example to verify that the proposed control strategy
efficiently works. The model used in the simulation is IEEJ EAST30 without PV plants
shown in the previous chapter.
Let us suppose that a fault happens at the fifth generator. The fault is modeled byan initial deviation of the angle of the generator. To reduce the influence made by thecontingency, we consider applying our method to the power system, where subsystems
are determined according to the set of the generators within the area. The global
controller KG generates the uniform actuation signal and sends it to all AVRs. On theother hand, the local controller KL,i sends an actuation signal only to the corresponding
AVRs.
Fig. 5.2 illustrated the trajectories of frequencies of the all generators in the grid.
The response in the case where no control is applied is shown by the red lines. Largeoscillation can be observed in this case of free response. The response in the case

79

Fig. 5.2 Responses of all generators’ frequencies, t ∈ [0, 10]. Red, green, and bluelines are obtained in the cases of no-control, retrofit control, proposed glocal control,
respectively.
where retrofit control is applied is shown by the green lines. It can be observed thatthe trajectories after the 5 second similarly behave as compared to the trajectories
of free responses. It is implied that incoherent dynamics can be suppressed by theretrofit controllers. Nevertheless, a uniform frequency deviation still remains and it
slowly converges to the setpoint. The response in the case where the proposed glocal
control is applied is shown by the blue lines. The glocal controllers are designed byheuristically ignoring the modeling error. The uniform frequency deviation, illustrated
by the green lines, is greatly reduced by the newly introduced global controller, eventhough the stability is not assured in theory for the modeling error. It should be notedthat the non-uniform dynamic is not stimulated by the global controller. These results
indicates the efficiency of the proposed glocal control for control of real power systems.

5.6

Chapter Summary

In this chapter, we have proposed a distributed design method for global and local
controllers for network systems. We have introduced a novel technique to represent the80
original system in a hierarchical one, called hierarchical representation. Thanks to thecascade structure of the hierarchical representation, it is enabled to distributedly design
glocal controllers. We have shown that decentralized rectifiers can be also designed ina distributed manner. Finally, simulation results to show the effectiveness of the glocal
control have been illustrated using the IEEJ EAST30 model.

Chapter 6
Conclusion
6.1

Summary of Results

This thesis has summarized the line of work on distributed design of controllers for
dynamical network systems. Summaries in individual chapters are as follows:First, in Chapter 2, a distributed design method of decentralized controllers basedon the retrofit control approach has been proposed. We have considered a large-scale
network system being stably operated where there are multiple operators with the aimof suppressing effects of disturbance injected into the network system by implementing
subcontrollers. For the network system, the distributed design problem has beenformulated by modeling the network system as an interconnected system composed ofa local system and an environment. The definition of retrofit controllers has been given
as the controllers that stabilize the interconnected system for any possible environment.
It has been shown that all retrofit controllers can be parameterized based on Youlaparameterization in the frequency domain. To investigate structure inside retrofit
controllers, we have introduced two kinds of retrofit controllers, output-rectifying
retrofit controllers and input-rectifying retrofit controllers. These controllers can bealso parameterized by Youla parameter, and moreover, it can be reduced to be aparameter without any constraint. Thanks to the unconstrained parameterization,
an optimal retrofit controller can be designed and consequently distributed design ofdecentralized controllers is achieved.
In Chapter 3, we have developed a retrofit controller design method without interconnection signal aiming at relaxing the assumption that the interconnection signal
can be measured. We have considered the case where interconnection signal is unavailable and derive a parameterization of all state-feedback retrofit controllers without
interconnection signal. It has been shown that a similar structure can be observed in82
state-feedback retrofit controllers to that of retrofit controllers with interconnection
signal.
In Chapter 4, we have considered PSS design based on retrofit control for frequency
control in power grids with renewable energy resources as an application of the proposed
distributed design of decentralized controllers. We have given a mathematical model of
EAST30 composed of synchronous machines, loads, PV power plants, non-unit buses,
and tie-line networks. For synchronous machines, Park model is employed and eachsynchronous machine consists of a synchronous generator, an excitation system with
AVR, and a turbine with governor. The loads and PV power plants are modeled as
constant loads and constant power sources, respectively. We have shown that thepower grid can be regarded as an interconnected system and formulated the problem
of distributed design of PSSs. To treat unstable local systems, we have developed aretrofit controller design method in the time-domain. Numerical simulation has beenshown to demonstrate the effectiveness of the proposed method.
As a generalization of the developed retrofit control, we have proposed a glocal
control approach for distributedly designing local and global controllers based onhierarchical representation in Chapter 5. We have derived necessary and sufficientconditions for existing of hierarchical representation for a network system. Based on
the hierarchical representation, we have proposed a glocal controller design methodwith rectifiers that preserve the hierarchical structure. Finally, the effectiveness of the
proposed method has been verified through a numerical example.

6.2

Future Works

We have shown that all output-rectifying retrofit controllers with interconnection andall state-feedback output-rectifying retrofit controllers without interconnection signal
have a structure inside the controller in Chapter 2 and 3. However, it is still unclear
whether output-feedback output-rectifying retrofit controllers have a similar structure
or not. One possible future direction is to generalize the result to the most general
setting.
In addition, in Chapter 5, we have developed a glocal control approach for a given
hierarchical representation. To search for a hierarchical representation for a given
system is an interesting future work.

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Acknowledgements
First and foremost, I fully thank my supervisor Jun-ichi Imura for his guidance andsupport and for providing me this precious study opportunity as a doctoral student
in his laboratory. All of my research topics in this thesis began by his insightful
suggestions, and carried out by his proper advice. I truly appreciate that he gave me
the opportunity to pursue my own study and other projects.
I am deeply grateful to Assistant Professor Takayuki Ishizaki for his enthusiastic
support, proper advice, and invaluable discussion that not only reveal the weakness
in my study but make my research of great achievement. Without his support andwidespread advice from mathematics to scientific writing and presentation, this work
does not exist.
I am also very grateful to Associate Professor Tomohisa Hayakawa, Professor
Kazuhiro Nakadai in Tokyo Institute of Technology, Assistant Professor Tomonori
Sadamoto in the University of Electro-Communications, and Dr. Takahiro Kawaguchi
in Tokyo Institute of Technology for valuable comments and suggestions.
I am indebted to my previous supervisors, Professor Emeritus Yutaka Yamamotoin Kyoto University and Professor Masaaki Nagahara in University of Kitakyushu whotaught me the fundamentals of scientific research.
I am very grateful to Noriko Sugimoto and Akiho Setoguchi for their support of mylife in the laboratory. Moreover, I thank current and past members of Imura, Hayakawa
and Nakadai laboratories for many great discussions and all the good times. Finally,my thanks for my family will never be enough.
Hampei Sasahara
Tokyo Institute of Technology

List of Publications
Journal Papers• Hampei Sasahara, Takayuki Ishizaki, Tomonori Sadamoto, Taisuke Masuta,
Yuzuru Ueda, Hideharu Sugihara, Nobuyuki Yamaguchi, Jun-ichi Imura, “Damping performance improvement for PV-integrated power grids via retrofit control.”
IFAC Control Engineering Practice, 2019. (accepted for publication)• Hampei Sasahara, Takayuki Ishizaki, Tomonori Sadamoto, Jun-ichi Imura, “Distributed design of local controllers for transient stabilization in power systems,”
Transactions of the Society of Instrument and Control Engineers, Vol. 53, No. 11,
pp. 608–617, 2017. (in Japanese)
Refereed International Conference Papers• Hampei Sasahara, Takayuki Ishizaki, Jun-ichi, Imura, “Parameterization of allstate-feedback retrofit controllers,” in Proc. IEEE Conference on Decision andControl, 2018.
• Hampei Sasahara, Takayuki Ishizaki, Masaki Inoue, Tomonori Sadamoto, Jun-ichi
Imura, “A characterization of all retrofit controllers,” in Proc. American Control
Conference, 2018.
• Hampei Sasahara, Takayuki Ishizaki, Tomonori Sadamoto, Jun-ichi Imura, Henrik Sandberg, Karl Henrik Johansson, “Glocal control for network systems via
hierarchical state-space expansion,” in Proc. IEEE Conference on Decision andControl, 2017.