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Restoration and Power Sharing in Microgrids in the Presence of Clock Drifts.
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Krishna, A, Schiffer, JF orcid.org/0000-0001-5639-4326 and Raisch, J (2018) A
Consensus-Based Control Law for Accurate Frequency Restoration and Power Sharing in
Microgrids in the Presence of Clock Drifts. In: 2018 European Control Conference (ECC).
2018 European Control Conference, 12-15 Jun 2018, Limassol, Cyprus. IEEE . ISBN
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A Consensus-Based Control Law for Accurate Frequency Restoration and
Power Sharing in Microgrids in the Presence of Clock Drifts*
Ajay Krishna
1
, Johannes Schiffer
2
and J
¨
org Raisch
1,3
Abstract— Clock drifts are a common phenomenon in dis-
tributed systems, such as microgrids (MGs). Unfortunately, if
not accounted for, the presence of clock drifts can hamper
accurate frequency restoration and power sharing in MGs. As
a consequence, we have proposed in [1] a distributed secondary
frequency control that ensures an accurate stationary control
performance in the presence of clock drifts. In the present work,
we extend the analysis in [1] by providing a tuning criterion
for the controller parameters that guarantees robust stability
of a given equilibrium point of the closed-loop dynamics with
respect to uncertain bounded clock drifts. Finally, our analysis
is validated via simulation.
I. INTRODUCTION
Electric power systems around the globe are currently
facing new changes and challenges which are mainly due to
the increasing presence of renewable energy sources (RESs).
At present, the electric power system contains a large number
of small units rather than a small number of large power
stations. These small units are usually equipped with RESs.
To interface RESs into the electric grid, power electronic
inverters are used. The physical characteristics of inverters
largely differ from the characteristics of conventional gener-
ators. Therefore, new and intelligent control concepts which
ensure stable and reliable power system operation are needed.
In this context, the concept of microgrids (MGs) is foreseen
as a promising solution [2]. A MG is a locally controllable
subset of a large power system. It consists of several RESs,
storage units and corresponding loads. MGs can typically
work in islanded or grid-tied mode [2]. In this paper, we are
interested in the former case.
As in any AC power system, frequency stability is a key
performance criterion in MGs. In inverter-dominated MGs,
so-called grid forming inverters (GFIs) are employed for this
task. A GFI is a voltage source inverter which is controlled
using pre-defined voltage and frequency values [2], [3]. In-
spired by conventional power systems, a hierarchical control
strategy is advocated in case of MGs [4], out of which, in this
paper, we are interested in distributed secondary frequency
control [1], [5]–[9] which uses local information as well as
neighboring information over a communication network to
ensure frequency restoration and power sharing (PS).
*The project leading to this manuscript has received funding from
the German Academic Exchange Service (DAAD) and the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No. 734832.
1
Fachgebiet Regelungssysteme, Technische Universit
¨
at Berlin, Germany.
{krishna, raisch}@control.tu-berlin.de
2
School of Electronic and Electrical Engineering, University of Leeds,
UK. j.schiffer@leeds.ac.uk
3
Max-Planck-Institut f
¨
ur Dynamik komplexer technischer Systeme,
Magdeburg, Germany.
In an inverter-dominated MG, each inverter typically has
only a local understanding of time, which leads to clock
inaccuracies [10]. In practice, clock drifts [11], [12] are
non-negligible phenomena in distributed MG control [1]. A
main reason for this is that in the presence of clock drifts,
the internal frequency of an inverter differs from its actual
electrical frequency [10], [13]. However, in most of the work
on distributed secondary frequency control, the effect of
clock drifts is not considered and it is assumed that both
the electrical and internal frequencies are identical. See for
example, the pinning control scheme [5], or the distributed
averaging integral (DAI) control [6]–[9]. In [1], we have
shown that the presence of clock drifts impairs accurate fre-
quency restoration and PS with the usual secondary control
schemes. Recently, the deteriorating effect of clock drifts on
secondary frequency control was reported also in [14]–[17].
As in sensor networks, time synchronization protocols
[18], [19] could potentially be used to address clock drift
issues. Yet, in the case of MGs, in order to implement these
time synchronization protocols, an additional time synchro-
nization control has to be designed and should typically
be activated before the primary and secondary controllers.
Adding such an additional control layer would increase the
overall complexity of the hierarchical MG control architec-
ture [4]. In [20], an angle droop control, with consensus
based frequency and power control to ensure PS in the
presence of clock drifts is proposed. Yet, the implementation
of [20] requires complete knowledge of phase angles, which
is often a restrictive assumption in practice. Furthermore, in
the related works [14]–[17] neither conditions for stability
nor conditions under which a given distributed frequency
control scheme leads to accurate PS in the presence of clock
drifts are provided.
Motivated by this, in [1], we have proposed an alternative
secondary control law, termed generalized distributed averag-
ing integral (GDAI) control, and provided a parametrization
of the control parameters, such that the synchronized elec-
trical frequency is the nominal frequency and, in addition,
PS is ensured in the presence of clock drifts. Moreover, the
GDAI control can be implemented without any additional
time synchronization protocol. In this paper, we extend the
work in [1] by providing a design criterion in the form of a
set of linear matrix inequalities (LMIs) which ensures that
the GDAI control renders asymptotic stability (AS) of the
closed-loop equilibrium point in the presence of clock drifts
and guarantees accurate frequency restoration and PS. Unlike
in [20], we do not linearize the electrical network, instead
we work with the non-linear MG model. Moreover, our
design criterion does not require knowledge of the operating
point. We use a Lyapunov function with classic kinetic and
potential energy terms to derive the design criterion [21],
[22]. Finally, we illustrate via simulation that with our design
criterion, the GDAI controller achieves accurate frequency
restoration, PS and local AS.
The paper is organized as follows. In Section II, we recall
some preliminaries of graph theory, introduce the MG model
and the GDAI control. In Section III, the design criterion
for the closed loop system in the presence of clock drifts
is presented. In Section IV, the design criterion is solved
numerically and the results are simulated for an exemplary
MG. Finally, we summarize our work and suggest some
future research directions in Section V.
II. PRELIMINARIES
We denote by I
n
the n × n identity matrix, by 0
n×m
the
n ×m matrix with all entries equal to zero, by 1
n
the vector
with all entries being equal to one and by 0
n
the zero vector.
The maximum eigenvalue of a square symmetric matrix F
is denoted by λ
max
(F ). The elements below the diagonal of
F is denoted by ∗. If F is positive (negative) definite, we
denote this by F > 0 (F < 0). If F is positive (negative)
semidefinite, we denote this by F ≥ 0 (F ≤ 0). Moreover,
A > B means that A − B > 0. Let x = col(x
i
) denote a
vector with entries x
i
, Y = diag(y
i
) a diagonal matrix with
diagonal entries y
i
and X = blkdiag(X
i
) a block-diagonal
matrix with block-diagonal matrix entries X
i
.
A. Algebraic graph theory
A weighted undirected graph [23], [24] of order n > 1 is
a triple G = (N , E, W) with set of vertices N = {1, . . . , n}.
Furthermore, E ⊆ [N ]
2
is the set of edges, where [N ]
2
represents the set of all two-element subsets of N and
W : E → R
>0
is a weight function. The entries of the
adjacency matrix A ∈ R
n×n
of G are a
ij
= a
ji
= w
l
> 0 if
{i, j} ∈ E where w
l
= w(i, j) = w(j, i) is the edge weight
and a
ij
= a
ji
= 0 otherwise. The set of neighboring nodes
of node i is given by N
i
= {j ∈ N | a
ij
6= 0}. The diagonal
degree matrix D ∈ R
n×n
is given by D = diag
P
j∈N
a
ij
.
A path is an ordered sequence of nodes such that any pair of
consecutive nodes in the sequence is connected by an edge.
The graph G is called connected if there exists a path between
every pair of distinct nodes. The Laplacian matrix L ∈ R
n×n
of an undirected graph is given by L = D−A. The Laplacian
matrix L is symmetric and positive semi-definite. If and only
if G is connected, L has a simple zero eigenvalue. Then, a
corresponding right eigenvector is 1
n
, i.e., L1
n
= 0
n
.
B. Primary-controlled MG model with clock drifts
We consider a Kron-reduced representation [25] of an
inverter-based MG and denote its set of network nodes by
N = {1, ..., n}, n > 1. The phase angle and voltage
magnitude at each bus i ∈ N are denoted by δ
i
: R
≥0
→ R,
respectively, V
i
: R
≥0
→ R
>0
. The electrical frequency at
the i-th node is denoted by ω
i
=
˙
δ
i
. As customary in sec-
ondary frequency control design, we assume that all voltage
amplitudes are constant and that the line admittances are
purely inductive [7], [25]. The latter assumption is generally
satisfied for MGs in which the inductive output impedance of
the converter filter and/or transformer dominates the resistive
part of the line impedances [26] and we only consider such
MGs. Thus, if there is a power line between nodes i ∈ N
and k ∈ N , then this is represented by a nonzero susceptance
B
ik
∈ R
<0
. Furthermore, the electrical network is assumed
to be connected and the set of neighboring nodes of the i-th
node is denoted by N
i
= {k ∈ N | B
ik
6= 0}.
Following [10], [13], we denote by µ
i
∈ R the constant
relative drift of the clock of the i-th unit, i ∈ N . In general,
|µ
i
| ≪ 1 is a small unknown parameter [10], [13]. Further-
more, it is convenient to introduce the internal frequency
¯ω
i
: R
≥0
→ R of the inverter at the i-th node which—under
the assumption of sufficiently fast sampling times—yields
the relation [10], [13] between the internal frequency ¯ω
i
and
the electrical frequency ω
i
as ¯ω
i
= (1 + µ
i
)ω
i
, ∀i ∈ N . In
the sequel, we refer to µ
i
∈ R as the clock drift factor or
simply clock drift.
Following standard practice, we assume that all units are
equipped with the usual primary frequency droop control [4].
Then the dynamics of the generation unit at the i-th node,
i ∈ N , is given by
(1 + µ
i
)
˙
δ
i
= (1 + µ
i
)ω
i
= ¯ω
i
,
(1 + µ
i
)M
i
˙
¯ω
i
= −D
i
(¯ω
i
− ω
d
) + P
d
i
− G
ii
V
2
i
+ u
i
− P
i
,
(II.1)
where ω
d
∈ R is the desired electrical frequency, P
d
i
∈ R is
the desired active power set point, G
ii
V
2
i
∈ R
≥0
represents
the constant power load at the i-th node, D
i
∈ R
>0
is the
inverse droop coefficient and M
i
= τ
P
i
D
i
is the virtual
inertia coefficient, where τ
P
i
∈ R
>0
is the time constant
of the low pass filter for the power measurement [27].
Furthermore, u
i
: R
≥0
→ R is the secondary control input.
The active power flow P
i
: R
n
→ R at the i-th node is [25]
1
P
i
=
X
k∈N
i
|B
ik
|V
i
V
k
sin(δ
i
− δ
k
). (II.2)
To derive a compact model representation of the MG, it is
convenient to introduce the matrices
D = diag(D
i
) ∈ R
n×n
>0
, M = diag(M
i
) ∈ R
n×n
>0
,
µ = diag(µ
i
) ∈ R
n×n
,
and the vectors
δ = col(δ
i
) ∈ R
n
, ω = col(ω
i
) ∈ R
n
, ¯ω = col(¯ω
i
) ∈ R
n
,
P
net
= col(P
d
i
− G
ii
V
2
i
) ∈ R
n
, u = col(u
i
) ∈ R
n
.
Also, we introduce the potential function U : R
n
→ R,
U(δ) = −
X
{i,k}∈[N ]
2
|B
ik
|V
i
V
k
cos(δ
i
− δ
k
).
Then, the dynamics (II.1), ∀i ∈ N , can be written as
(I
n
+ µ)
˙
δ = ¯ω,
(I
n
+ µ)M
˙
¯ω = −D(¯ω − 1
n
ω
d
) + P
net
+ u − ∇U (δ),
(II.3)
compactly. Observe that
dU(δ)
dt
= ∇U
⊤
(δ)ω = ∇U
⊤
(δ)(I + µ)
−1
¯ω,
and due to symmetry of the power flows P
i
, 1
⊤
n
∇U(δ) = 0.
1
For notational simplicity, time arguments of all signals are omitted.
C. Generalized Averaging Integral Secondary control
In general, P
net
in (II.3) is non-zero because the loads
G
ii
V
2
i
are usually unknown. Moreover, if the overall power
balance is non-zero, then the steady state frequencies of the
droop controlled MG (II.3) deviate from the nominal value
ω
d
. This steady state frequency error should be brought to
zero using a secondary control law. In this paper, we focus on
distributed secondary frequency control. Note that usually the
internal frequency ¯ω
i
is employed in distributed secondary
frequency control [5], [6], [8], [9]. At first glance, this has
the advantage that no additional frequency measurement is
needed. However, it has been shown in [1] that the, generally
unavoidable, presence of clock drifts also leads to non-
negligible stationary frequency deviations when using any
of the aforementioned control schemes. Motivated by this,
we have proposed the following GDAI control in [1]
u = p, (I
n
+ µ) ˙p = −
(B + CL
C
)(¯ω − 1
n
ω
d
) + DL
C
p
,
(II.4)
where B ∈ R
n×n
, C ∈ R
n×n
and D ∈ R
n×n
>0
are diagonal
controller matrices and L
C
∈ R
n×n
≥0
is the Laplacian matrix
representing the communication network. For the subsequent
analysis, it is convenient to introduce the notion below.
Definition 2.1: The closed loop system (II.3), (II.4) admits
a synchronized motion if it has a solution for all t ≥ 0 of
the form
δ
s
(t) = δ
s
0
+ ω
s
t, ω
s
= ω
∗
1
n
,
where ω
∗
∈ R is the synchronized electrical frequency and
δ
s
0
∈ R
n
such that
|δ
s
0,i
− δ
s
0,k
| <
π
2
∀i ∈ N , ∀k ∈ N
i
.
The expression for ω
∗
is given by [1]
ω
∗
=
1
T
n
D
−1
B1
n
1
T
n
D
−1
(B + CL
C
)(I
n
+ µ)1
n
ω
d
. (II.5)
The matrix B is commonly called the pinning gain matrix,
e.g. [5]. In the following, we define the clock of one of the
units in the network as master clock, say the k-th unit, k ≥ 1.
Then, µ
k
= 0 and the drifts µ
i
, i 6= k of all other clocks in
the MG are expressed with respect to the master clock at k-th
unit. In this scenario, it has been shown in [1] that selecting
Bµ = 0
n×n
, C = −D, (II.6)
in (II.4) guarantees that ω
∗
= ω
d
in (II.5) together with
active PS in the presence of clock drifts. Achieving both of
these control objectives is, in general, not possible with [5]
and the standard DAI control [6], [8], [9] in the presence of
clock drifts, see [1] for more details.
D. Closed-Loop System
Combining (II.3) with (II.4) and using (II.6) yields the
closed-loop dynamics
(I
n
+ µ)
˙
δ = ¯ω,
(I
n
+ µ)M
˙
¯ω = −D(¯ω − 1
n
ω
d
) − ∇U (δ) + P
net
+ p,
(I
n
+ µ) ˙p = −(B − DL
C
)(¯ω − 1
n
ω
d
) − DL
C
p.
(II.7)
III. ROBUST GDAI CONTROL DESIGN
In this section, a tuning criterion is derived that ensures
robust stability of the closed-loop MG dynamics (II.7) in the
presence of clock drifts.
A. Error States and Problem Statement
We make the following standard assumption.
Assumption 3.1: The closed-loop system (II.7) possesses
a synchronized motion.
As the power flow equation (II.2) only depends on angle
differences, following [26] we choose an arbitrary node, say
node n and express all angles relative to that node, i.e.,
θ = R
⊤
δ, θ ∈ R
n−1
, R =
I
n−1
−1
⊤
n−1
.
Then, with Assumption 3.1, we introduce the error states
˜ω = ¯ω − ¯ω
∗
= ¯ω − ( I
n
+ µ)
−1
1
n
ω
d
,
˜
θ = θ − θ
∗
, ˜p = p − p
∗
, x = col
˜
θ, ˜ω, ˜p
.
The resulting error dynamics of the system (II.7) is given by
˙
˜
θ = R
⊤
(I
n
+ µ)
−1
˜ω,
(I
n
+ µ)M
˙
˜ω = −D˜ω − R(∇U(
˜
θ + θ
∗
) − ∇U (θ
∗
)) + ˜p,
(I
n
+ µ)
˙
˜p = (−B + DL
C
)˜ω − DL
C
˜p,
(III.1)
we define x = col(
˜
θ, ˜ω, ˜p) ∈ R
3n−1
and x
∗
= 0
3n−1
is an
equilibrium point of (III.1). Note that AS of x
∗
= 0
3n−1
implies AS of the synchronized motion from Definition 2.1
in system (II.7) up to a uniform shift of all angles [26]. As
outlined in [10], [13] for the purpose of secondary frequency
control, it is reasonable to assume that the clock drifts are
bounded. This is formalized in the assumption below.
Assumption 3.2: kµk
2
≤ ǫ, 0 ≤ ǫ < 1.
We are interested in the following problem.
Problem 3.3: Consider the system (III.1) with Assump-
tion 3.1. Determine the matrices B, D and L
C
, such that
AS of x
∗
is guaranteed for all µ satisfying Assumption 3.2.
B. Main result
For the presentation of our main result, it is convenient to
define the matrices
T =
"
T
11
1
2
−I
n
− σD
−1
1
n
1
⊤
n
D +
˜
B − L
C
∗ T
22
#
,
ˆ
T
2
=
σM1
n
1
⊤
n
˜
B σD
−1
1
n
1
⊤
n
D
0
n×n
−σD
−1
1
n
1
⊤
n
,
(III.2)
with σ > 0,
˜
B = D
−1
B ≥ 0 and
T
11
= D −
σ
2
M1
n
1
⊤
n
˜
B +
˜
B1
n
1
⊤
n
M
,
T
22
= L
C
+
σ
2
D
−1
1
n
1
⊤
n
+ 1
n
1
⊤
n
D
−1
.
Furthermore, since µ is a diagonal matrix, with Assump-
tion 3.2 we have that
kµ(I
n
+ µ)
−1
k
2
≤ g
1
(ǫ), g
1
(ǫ) =
ǫ
1−ǫ
> 0,
k(µ
2
+ 2µ)(I
n
+ µ)
−2
k
2
≤ g
2
(ǫ), g
2
(ǫ) =
ǫ
2
+2ǫ
(1−ǫ)
2
> 0.
(III.3)
Our main result is as follows.
Proposition 3.4: Consider the system (III.1) with As-
sumption 3.1. Recall g
1
(ǫ) and g
2
(ǫ) defined in (III.3).
Suppose that there exist σ > 0, ζ > 0 such that
H
nom
=
M −σM1
n
1
⊤
n
D
−1
∗ D
−1
>
g
2
(ǫ)M 0
n×n
0
n×n
g
1
(ǫ)D
−1
,
(III.4)
and
T >
ǫζ + g
1
(ǫ)
p
λ
max
(D
2
) + 1
I
2n
,
0 ≥
−ζI
2n
ˆ
T
2
∗ −ζI
2n
,
(III.5)
where T and
ˆ
T
2
are defined in (III.2). Then, local AS of
x
∗
= 0
3n−1
is guaranteed for all unknown clock drift factors
satisfying Assumption 3.2.
Proof: Consider the Lyapunov function candidate
V =
1
2
˜ω
⊤
M ˜ω + U (
˜
θ + θ
∗
) − ∇U (θ
∗
)
⊤
˜
θ
+
1
2
˜p
⊤
D
−1
(I
n
+ µ)˜p
− σ˜p
⊤
(I
n
+ µ)D
−1
1
n
1
⊤
n
M(I
n
+ µ)˜ω,
(III.6)
where σ > 0 is a design parameter. The Lyapunov function
V contains kinetic and potential energy terms ˜ω
⊤
M ˜ω, re-
spectively U(
˜
θ) [21], a quadratic term in secondary control
input ˜p and a cross term between ˜ω and ˜p which allows us to
ensure that V is decreasing along the trajectories of (III.1).
First, we will show that V is indeed positive definite. Note
that ∇
x
V
x
∗
= 0
3n−1
. This shows that x
∗
is a critical point
of V . Moreover, the Hessian of V at x
∗
is given by
∇
2
x
V |
x∗
=
∇
2
U(θ
∗
) 0
(n−1)×n
0
(n−1)×n
∗ M ∇
2
V |
(2,3)
∗ ∗ D
−1
(I
n
+ µ)
, (III.7)
with ∇
2
V |
(2,3)
= −σ(I
n
+ µ)M1
n
1
⊤
n
D
−1
(I
n
+ µ). Note
that the matrix ∇
2
U(θ
∗
) > 0 [26]. Therefore, the Hessian
∇
2
x
V |
x∗
is positive definite if and only if
M −σ(I
n
+ µ)M1
n
1
⊤
n
D
−1
(I
n
+ µ)
∗ D
−1
(I
n
+ µ)
> 0. (III.8)
By performing a congruence transformation using the pos-
itive definite matrix S = blkdiag
(I
n
+ µ)
−1
, (I
n
+ µ)
−1
and by invoking Sylvester’s law of inertia [28], we see that
the matrix on the left hand side of (III.8) is positive definite
if and only if the following matrix inequality is satisfied
(I
n
+ µ)
−2
M −σM1
n
1
⊤
n
D
−1
∗ (I
n
+ µ)
−1
D
−1
> 0. (III.9)
The inequality (III.9) can be written as
H
nom
−
(µ
2
+ 2µ)(I
n
+ µ)
−2
M 0
n×n
0
n×n
µ(I
n
+ µ)
−1
D
−1
> 0,
where H
nom
is defined in (III.4). Furthermore, since µ , M
and D are all diagonal matrices, we have that
(µ
2
+ 2µ)(I
n
+ µ)
−2
M 0
n×n
0
n×n
µ(I
n
+ µ)
−1
D
−1
≤
g
2
(ǫ)M 0
n×n
0
n×n
g
1
(ǫ)D
−1
,
where g
1
(ǫ) and g
2
(ǫ) are defined in (III.3). Consequently,
under the standing assumptions, see (III.4), ∇
2
x
V |
x∗
>
0, confirming the positive definiteness of V . Note that
∇
x
V
x
∗
= 0
3n−1
and ∇
2
x
V |
x∗
> 0 implies that x
∗
is a
strict local minimum of V [29].
Next, we calculate the time derivative of V along the
solutions of (III.1), which yields
˙
V (η) = −˜ω
⊤
(I
n
+ µ)
−1
D˜ω + ˜ω
⊤
(I
n
+ µ)
−1
˜p − ˜p
⊤
D
−1
B˜ω
+ σ ˜p
⊤
(I
n
+ µ)D
−1
1
n
1
⊤
n
D˜ω − σ ˜p
⊤
(I
n
+ µ)D
−1
1
n
1
⊤
n
˜p
+ ˜p
⊤
L
C
˜ω − ˜p
⊤
L
C
˜p + σ˜ω
⊤
(I
n
+ µ)M 1
n
1
⊤
n
D
−1
B˜ω,
= −η
⊤
T
11
T
12
∗ T
22
η = −η
⊤
Tη,
(III.10)
where η = col(˜ω, ˜p) and
T
11
= (I
n
+ µ)
−1
D
−
σ
2
(I
n
+ µ)M 1
n
1
⊤
n
D
−1
B + D
−1
B1
n
1
⊤
n
M(I
n
+ µ)
,
T
22
= L
C
+
σ
2
(I
n
+ µ)D
−1
1
n
1
⊤
n
+ 1
n
1
⊤
n
D
−1
(I
n
+ µ)
,
T
12
=
1
2
−(I
n
+ µ)
−1
− σ(I
n
+ µ)D
−1
1
n
1
⊤
n
D + D
−1
B − L
C
.
Note that the entries of the matrix T in (III.10) are uncertain,
because the clock drift matrix µ is uncertain. Hence, to obtain
verifiable conditions that ensure T > 0 and, thus,
˙
V (η) is
negative definite, we note that T can be decomposed as
T = T −
1
2
Γ
1
ˆ
T
1
+
ˆ
T
⊤
1
Γ
1
−
1
2
Γ
2
ˆ
T
2
+
ˆ
T
⊤
2
Γ
2
, (III.11)
where
ˆ
T
2
is defined in (III.2) and
Γ
1
= blkdiag
µ(I
n
+ µ)
−1
, µ(I
n
+ µ)
−1
,
Γ
2
= blkdiag (µ, µ) ,
ˆ
T
1
=
D −I
n
0
n×n
0
n×n
.
(III.12)
For any matrices A ∈ R
n×n
and B ∈ R
n×n
, it holds that
AB + B
⊤
A
⊤
≤ 2kAk
2
kBk
2
I
n
.
Therefore from (III.11), we have that
T ≥ T −
k
ˆ
T
1
k
2
kΓ
1
k
2
+ k
ˆ
T
2
k
2
kΓ
2
k
2
I
2n
.
Assumption 3.2 together with (III.3), implies that
kΓ
1
k
2
≤ g
1
(ǫ), kΓ
2
k
2
≤ ǫ,
where Γ
1
and Γ
2
are defined in (III.12). Therefore,
T ≥ T −
g
1
(ǫ)k
ˆ
T
1
k
2
+ ǫk
ˆ
T
2
k
2
I
2n
. (III.13)
From (III.12), we have that
k
ˆ
T
1
k
2
=
q
λ
max
(
ˆ
T
1
ˆ
T
⊤
1
) =
p
λ
max
(D
2
) + 1.
Turning to
ˆ
T
2
,
k
ˆ
T
2
k
2
=
q
λ
max
(
ˆ
T
2
ˆ
T
⊤
2
) ≤ ζ ⇔ λ
max
(
ˆ
T
2
ˆ
T
⊤
2
) ≤ ζ
2
,
⇔
ˆ
T
2
ˆ
T
⊤
2
≤ ζ
2
I
2n
,
⇔
1
ζ
ˆ
T
2
ˆ
T
⊤
2
− ζI
2n
≤ 0,
