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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 64, NO. 11, NOVEMBER 2017 1327
A Passivity-Preserving Frequency-Weighted
Model Order Reduction Technique
Umair Zulfiqar, Waseem Tariq, Li Li, and Muwahida Liaquat
Abstract—Frequency-weighted model order reduction techniques aim to yield a reduced order model whose output matches
that of the original system in the emphasized frequency region.
However, passivity of the original system is only known to be
preserved in the single-sided weighted case. A frequency-weighted
model order reduction technique is proposed, which guarantees
the passive reduced models in the double-sided weighted case.
A set of easily computable error bound expressions are also
presented.
Index Terms—Error bound, frequency-weighted Gramians,
model order reduction, passivity.
I. INTRODUCTION
MODEL order reduction (MOR) has been the attention of
researchers for the past few decades. The aim of MOR
is to find a fairly accurate lower order approximate model
which retains the essential properties of the original system
like stability, passivity and input-output behaviour [1]–[5].
Truncated balanced realization (TBR) [6] is one of the most
popular MOR techniques due to its accuracy, preservation of
stability and easily computable a priori error bound.
In the context of MOR of RLC networks, integrated circuits, control systems and interconnected circuits, passivity is
an important property to be preserved because non-passive
reduced order model (ROM) may yield nonphysical behaviour
by generating energy at high frequencies and resulting in
erratic time-domain behaviours. Passivity implies stability but
the opposite is not true. Phillips et al. [7] extended TBR [6]
to preserve passivity of the original system and also presented
the corresponding error bound expressions.
In many practical applications, it is desirable that the
output of ROM matches that of the original system in
some specified frequency region [8]. Enns [9] presented a
frequency-weighted generalization of TBR [6] for that purpose. However, stability is only guaranteed in the case of
single-sided frequency weighting with no a priori error bound
Manuscript received November 14, 2016; revised January 18, 2017;
accepted March 18, 2017. Date of publication March 21, 2017; date of current
version November 1, 2017. This brief was recommended by Associate Editor
T. Fernando. (Corresponding author: Umair Zulfiqar.)
U. Zulfiqar, W. Tariq, and M. Liaquat are with the Department of
Electrical Engineering, College of Electrical and Mechanical Engineering,
National University of Sciences and Technology, Islamabad 44000, Pakistan
(e-mail: umairzulfiqar79@ee.ceme.edu.pk; waseemtariq79@ee.ceme.edu.pk;
muwahida@ee.ceme.edu.pk).
L. Li is with the School of Electrical, Mechanical and Mechatronic Systems,
University of Technology Sydney, Sydney, NSW 2007, Australia (e-mail:
li.li@uts.edu.au).
Digital Object Identifier 10.1109/TCSII.2017.2685440
provided. Several modifications exist in [10]–[13] to ensure
stability in the double-sided weighted case and error bounds
are also derived. These techniques [10]–[13], however, do not
guarantee passivity of the ROM.
Heydari and Pedram [14] proposed a frequency-weighted
generalization of Phillips et al.’s technique [7]. Similarly,
Enns’ technique [9] and its modifications [10], [11] are
generalized in [15] which claimed to yield the guaranteed
passive ROMs. However, it is pointed in [16] that these
techniques [14], [15] guarantee the passivity only in the singlesided weighted case. Many approximation criteria in MOR are
double-sided weighted criteria, particularly, when the model to
be reduced is part of a closed-loop system and preservation
of the closed-loop behaviour is important [17]. Frequencyweighted MOR (FWMOR) techniques are used to satisfy
these criteria and preservation of passivity in the double-sided
weighted case is, therefore, critical.
To the authors’ best knowledge, there is no FWMOR technique in the literature so far which guarantees the passivity
in the double-sided weighted case. In this brief, an FWMOR
technique is proposed which preserves passivity of the original system both in the single and double-sided weighted
cases. A set of easily computable error bound expressions are
also derived. Numerical examples are presented to show the
efficacy of proposed technique.
II. BACKGROUND
Consider an nth order positive-real linear time invariant
system
G(s) = C(sI − A)
−1B + D
where {A, B,C, D} is its minimal state-space realization, A ∈
Rn×n, B ∈ Rn×m,C ∈ Rm×n and D ∈ Rm×m.
A. Phillips et al.’s Technique [7]
The controllability Gramian-like matrix Pa and the observability Gramian-like matrix Qa of the system {A, B,C, D} are
the solution of following Lur’e equations:
APa + PaAT = −KcKT
c (1)
PaCT − B = −KcJT
c JcJT
c = D + DT (2)
ATQa + QaA = −KT
o Ko (3)
QaB − CT = −KT
o Jo JT
o Jo = D + DT (4)
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