Published at : 31 Jan 2025
Volume : IJtech
Vol 16, No 1 (2025)
DOI : https://doi.org/10.14716/ijtech.v16i1.7180
Huy-Du Dao | Faculty of Electronics, Thai Nguyen University of Technology, Thai Nguyen 251750, Vietnam |
Thanh-Tung Nguyen | Faculty of Engineering and Technology, Thai Nguyen University of Information and Communication Technology, Thai Nguyen 250000, Vietnam |
Ngoc-Kien Vu | Research Development Institute of Advanced Industrial Technology (RIAT), Thai Nguyen University of Technology, Thai Nguyen 251750, Vietnam |
Van-Ta Hoang | College of Technology and Trade, Thai Nguyen 250000, Vietnam |
Hong-Quang Nguyen | Faculty of Mechanical, Electrical, Electronics Technology, Thai Nguyen University of Technology, Thai Nguyen 251750, Vietnam |
This study focuses on Model Order Reduction (MOR)
to optimize such systems' simulation and analysis capabilities in the context
of increasingly complex electrical and electronic systems, coupled with
computational and processing resource limitations. This paper proposes the
Mixed Balanced Truncation (MBT) algorithm, which combines the strengths of
Balanced Truncation (BT) and Positive-Real Balanced Truncation (PRBT) while
addressing their respective limitations. The MBT algorithm is developed based
on Lyapunov and Riccati equations, ensuring the stability and passivity of the
reduced-order system. The proposed method is validated through large-scale
electrical circuit systems, using RLC network models as illustrative examples.
The results demonstrate that MBT achieves effective order reduction with
minimal error while reducing computational costs. The main contributions of this
work in developing the new reduction algorithm include the introduction of a
novel definition of mixed balanced systems and theoretical advancements through
the development of theorems, lemmas, and corollaries accompanied by rigorous
mathematical proofs. This study makes significant theoretical contributions and
provides practical solutions for designing, modeling, and reducing the
complexity of electrical and electronic systems, particularly passive linear
systems in general.
Computational Efficiency; Descriptor Systems; Electrical Circuit Simulation; Mixed Balanced Truncation; Model Order Reduction
In circuit simulation, Modified Nodal Analysis (MNA) is a
widely used method for constructing mathematical models of circuit behavior (Choupanzadeh and Zadehgol, 2023; Pavan and Temes, 2023; Hao
and Shi, 2022; Günther et al., 2005). This technique represents the input-output
relationship of a circuit as a linear descriptor
system, expressed by the following equation (1).
|
(1) |
where
Remark
1. The system described by equation (1)
exhibits several key characteristics and requirements (these properties ensure
the system meets the demands for simulation, analysis, and the application of
model reduction algorithms):
-
The system is minimal,
ensuring no redundant states.
-
The dynamics are stable and
passive, indicating that the system does not generate energy and all
eigenvalues of the pencil matrix pair (E, A) have non-positive real
parts.
-
The initial conditions are
such that the state variables, inputs, and outputs are all zero.
-
Matrices A and E are nonsingular, with ranks equal to n, and matrices A and E-1A are stable.
-
The matrix D satisfies the condition D + DT ? 0, ensuring that the system's output matrix is
positive semi-definite.
Dimensionality Reduction (DR) or Model Order Reduction
(MOR) is a crucial technique widely applied in mathematical modeling across
various fields, including electrical and electronic systems (Benner et al.,
2021; Benner et al., 2020; Fortuna et al., 2012; Schilders et al., 2008).
The primary goal of DR and MOR is to simplify a complex, high-dimensional model
by replacing it with a lower-dimensional model while preserving the system's
essential physical properties and dynamic behavior. This simplification
significantly reduces computational complexity, enabling faster computations
required for real-time simulations. Furthermore, it reduces computational
workload and storage requirements, enhancing hardware performance and cost
efficiency, especially in resource-constrained environment.
In
large-scale circuit simulations, many model reduction algorithms are applied in
various critical applications. Among them, methods such as Krylov subspace (Freund, 2022; Freund,
2000), Rational Krylov (Ali et al., 2019), Moment Matching (Benner and Feng, 2021;
Prajapati and Prasad, 2020), Asymptotic Waveform Evaluation (Jiang and Yang,
2021), Lanczos technique (Wittig et al., 2002), Arnoldi iteration (Song et al., 2017;
Jiang and Xiao, 2015), Proper Orthogonal Decomposition (Gräßle et al.,
2020; Manthey et al., 2019), Singular Value Decomposition (Younes et al.,
2021), Principal Component Analysis (Anaparthi et al., 2005), Padé
approximation (Singh et al., 2008), Singular Perturbation (Khan et al., 2019; Huisinga and
Hofmann, 2018), Matrix Interpolation (Kassis et al., 2016; Samuel et al.,
2014), frequency weighting balance truncation (Floros et al., 2019; Rydel and
Stanis?awski, 2018), time weighting balance truncation (König and Freitag,
2023), and others stand prominent (Kumar and Ezhilarasi, 2023a; Gugercin and
Antoulas, 2004).
During
the exploration of foundational methodologies prior to developing a new model
order reduction algorithm, the research team focused particularly on two
techniques: Balanced Truncation (BT) (Axelou et al., 2023; Kumar and Ezhilarasi, 2023b; Hossain
and Trenn, 2023; Suman and Kumar, 2021; Grussler et al., 2021; Antoulas, 2005;
Mehrmann and Stykel, 2005) and Positive-Real Balanced Truncation (PRBT)
(Poort et
al., 2023; Breiten and Unger, 2022; Zulfiqar et al., 2018; Benner and Stykel, 2017;
Reis and Stykel, 2010; Reis and Stykel, 2010; Tan and He, 2007; Tan and He,
2007). These methods were prioritized due to their ability to preserve
the physical properties of the original system, specifically stability (for BT)
and passivity (for PRBT). Both BT and PRBT ensure that reduced-order models
retain these crucial physical characteristics, which are essential for circuit
simulations.
Balanced
Truncation (BT), a classical approach to MOR, was first introduced by Moore in
1981. This method has been extensively studied, refined, and applied across
various applications (Axelou et al., 2023; Kumar and Ezhilarasi, 2023b; Hossain and
Trenn, 2023; Suman and Kumar, 2021; Grussler et al., 2021; Antoulas, 2005; Mehrmann
and Stykel, 2005). BT operates by balancing the controllability and
observability Gramians of the system and then truncating states with small
singular values. This technique ensures system stability is preserved and
achieves minimal reduction error, particularly in cases of moderate-order
reduction. However, a notable limitation of BT is its inability to maintain the
passivity of the original system, which is critical in practical applications,
such as ensuring that electrical circuits do not generate energy (Breiten and Unger,
2022).
To
address the limitations of BT, the Positive-Real Balanced Truncation (PRBT)
method was developed for positive-real systems (Reis and Stykel, 2010; Tan and He, 2007).
PRBT enables model order reduction while preserving the system’s passivity, a
property that ensures the system cannot generate or amplify energy. Numerous
studies have explored improvements and applications of PRBT, highlighting its
significance in various fields (Poort et al., 2023; Breiten and Unger, 2022; Zulfiqar et
al., 2018; Benner and Stykel, 2017). Although PRBT preserves essential
physical properties such as stability and passivity, it often leads to larger
reduction errors compared to BT and involves solving more complex problems,
resulting in higher computational costs.
Several
hybrid methods have been proposed to combine the strengths of BT and PRBT while
mitigating their respective weaknesses (Salehi et al., 2022; Salehi et al., 2021a; 2021b;
2021c; Zulfiqar et al., 2020; Lindmark and Altafini, 2017; Zulfiqar et al.,
2017; Unneland et al., 2007a; 2007b; Phillips et al., 2002). These
methods often involve complex balancing techniques or employ mixed Gramians.
These methods often rely on intricate balancing techniques or the use of mixed
Gramians. However, their applicability is typically restricted to standard
linear systems, rendering them unsuitable for direct application to linear
descriptor systems, which are frequently encountered in circuit analysis
models. Furthermore, methods described in studies such as (Salehi,
Karimaghaee and Khooban, 2021a; Salehi et al., 2022; Salehi, Karimaghaee, and
Khooban, 2021b) require solving two Riccati equations, which adds
computational complexity and cost.
To
overcome these challenges, we propose a novel algorithm named Mixed Balanced
Truncation (MBT), specifically designed to reduce the order of continuous-time
descriptor systems in the circuits model. MBT leverages the advantages of both
BT and PRBT by incorporating techniques that ensure the system remains stable
and passive while minimizing computational costs and reducing errors. This
novel approach addresses the limitations of previous methods and provides a
more efficient solution for DR in practical applications.
The primary contributions of this paper include a comprehensive definition of the proposed methods, three theorems that establish the theoretical foundation, two lemmas, two corollaries, and a new algorithm. The effectiveness and applicability of the proposed MBT method are demonstrated through illustration examples and simulations. This study advances the current body of literature by providing new insights and practical solutions for model order reduction in large-scale electronic circuit simulations, representing a significant step forward in this research field. This research contributes to the theoretical understanding of MOR. It offers a practical algorithm that can be applied to improve the efficiency and performance of electronic and electrical systems. The proposed MBT method paves the way for more effective and reliable simulations, crucial for designing and analyzing modern complex systems.
Preliminaries
2.1. Balanced
Truncation (BT) for reducing model order
The Balanced Truncation (BT) algorithm is
constructed on the principle of balancing the Grammians of the system. This
involves using a non-singular transformation matrix to equalize and diagonalize
the controllability and observability of Grammians. The reduced-order model is
then obtained by eliminating the modes associated with minor Hankel singular
values, which represent low-energy modes with minimal influence on the system's
behavior. The implementation details of the BT algorithm are presented in (Axelou et al.,
2023; Kumar and Ezhilarasi, 2023b; Hossain and Trenn, 2023; Suman and Kumar,
2021; Grussler et al., 2021; Mehrmann and Stykel, 2005; Antoulas, 2005).
Theorem 1 (Antoulas,
2005). If the system described by Equation (1) is stable, then the
matrices Kc (controllability Gramian) and Ko
(observability Gramian) are symmetric and positive definite. These matrices
satisfy the following Lyapunov equations (2) and (3).
|
(2) |
|
(3) |
2.2. Positive-Real Balanced
Truncation (PRBT) for reducing model order
The Positive-Real Balanced Truncation (PRBT) algorithm extends the
principles of BT to address passive systems specifically. In this technique,
the matrices Jc (control Gramian) and Jo
(observation Gramian) are computed by solving two Riccati equations. The
implementation details of the PRBT algorithm are presented in (Poort et al.,
2023; Breiten and Unger, 2022; Zulfiqar et al., 2018; Benner and Stykel, 2017;
Reis and Stykel, 2010; Reis and Stykel, 2010; Tan and He, 2007; Tan and He,
2007).
Theorem 2 (Reis
and Stykel, 2010; Tan and He, 2007). A system described by equation (1)
exhibits passivity if and only if its transfer function G(s) is of the
positive-real type. This condition is met if there exist matrices Jc and Jo that satisfy the
following Riccati equations (4) and (5).
|
(4) |
|
(5) |
These equations ensure that the system maintains passivity while reducing
its order, effectively preserving the original system's essential
characteristics.
Reduction of
Model Order Using Mixed Balanced Truncation
The Mixed Balanced Truncation algorithm is
utilized to reduce the model order of mixed-balanced systems. Therefore, it is
first necessary to determine whether the system in question conforms to this
balanced form. A system is said to satisfy the mixed balanced property if it
fulfills the criteria specified in Definition 1.
Definition 1. A linear
descriptor system described by equation (1) satisfying Remark 1 is termed a
mixed-balanced system if it meets the following conditions (6) or (7).
|
(6) |
|
(7) |
where Kc and Jo satisfy Equations (2) and (5), and Jc and Ko satisfy Equations
(3) and (4). are the Hankel singular values of the
mixed-balanced system with
.
Remark 2. If the system described by equation (1) does
not satisfy the criteria defined in Definition 1, then it is possible to
convert this system into a mixed-balanced system using Theorem 3.
Theorem 3. Consider the system described by equation
(1) satisfying the conditions outlined in Remark 1. A non-singular
transformation always exists via a transformation matrix Tz, such as equation (8) or (9).
|
(8) |
|
(9) |
Proof of Theorem 3. By applying the
Cholesky decomposition to Kc and Jo,
followed by performing Singular Value Decomposition on the matrix , being the Cholesky factors, we then calculate Tz
and its inverse. This results in equations (10) and (11).
|
(10) |
|
(11) |
Lemma 1. For a system described by equation (1) satisfying
Remark 1, the eigenvalues of the matrices KcJo or JcKo are positive and remain invariant under
non-singular transformations facilitated by the matrix Tz.
Proof
of Lemma 1. By performing the
diagonalization of the matrix product KcJo, we derive , where ? is a diagonal matrix containing the eigenvalues ?i of KcJo; Tz represents a matrix with eigenvectors of KcJo as its columns. Additionally, we have equation
(12).
|
(12) |
Thus, we can infer that , where
are
the Hankel singular values of the mixed-balanced system, satisfying
. Therefore, Lemma
1 is proven.
Lemma 2. For a system
described by equation (1) that meets the conditions in Remark 1, achieving a
mixed-balanced state through a non-singular transformation using the matrix Tz
is always possible, resulting in the new system matrices described by equation
(13).
|
(13) |
Proof of Lemma 2. When
equations (8) and (13) are substituted into equations (2) and (5), the updated
system is described by equations (14) and (15).
|
(14) |
|
(15) |
These new
equations (14) and (15) possess solutions that meet the criteria of Definition
1. Therefore, by employing the equivalent transformation with the non-singular
matrix Tz, the original system converts into the
mixed-balanced system, thereby establishing the validity of Lemma 2.
Algorithm 1. Reduce Model Order Using Mixed Balanced
Truncation |
|
Input: The dynamical system G(s) described by equation 1 satisfying the
conditions of Remark 1. Output: Reduced-order system described by the reduced matrices: |
|
Equivalent transformation of the original
system to a mixed-balanced system: 1. Compute Kc
and Jo from equations (2) and (5). 2. Perform Cholesky
decomposition on Kc and Jo
as equations (16) and (17). |
|
|
(16) |
|
(17) |
where
P and Q are invertible lower triangular matrices. |
|
3. Decompose the
singular values of the product |
|
|
(18) |
where
U and V are orthogonal matrices, and XB
is a diagonal matrix containing singular values. |
|
4. Calculate the conversion matrix Tz and
inverse according to equations (19) and (20). |
|
|
(19) |
|
(20) |
5. Calculate the updated system
matrices of the mixed-balanced system using the conversion equation (13). |
|
Reduce Model Order Using Mixed
Balanced Truncation: |
|
6. Choose the intended reduced
dimension r where 0 < r < n. |
|
7. Compute the projection
matrices as equations (26) and (27). |
|
|
(21) |
|
(22) |
8.
Calculate the matrices of the reduced order system as in the equations in
(23). |
|
|
(23) |
Remark 3. In this algorithm, we utilize the Gramians Kc and Jo. Alternatively, we can use the Gramians Jc and Ko owing to the balanced and symmetric nature of the
system.
Corollary
1. The mixed-balanced system
obtained from Algorithm 1 (from Step 1 to Step 5) retains the properties
described in Remark 1. The control matrix XBc and the observer matrix XBo of the mixed-balanced system are symmetric,
positive definite diagonal matrices, as in expression (24).
|
(24) |
Proof of corollary 1. To verify the stability and passivity of the
resulting mixed-balanced system, we solve the Lyapunov equation for the new
control matrix XBc and the Riccati equation for the new
observation matrix XBo, as specified in equations (25) and (26),
respectively.
|
(25) |
|
(26) |
where:
|
(27) |
|
(28) |
The
matrices XBc and XBo exhibit diagonal symmetry and positive definiteness, confirming
the conditions specified in Theorem 1 and Theorem 2. Therefore, the resulting
mixed-balanced system shows stable and passive behavior
Theorem 4: The reduced-order system obtained from
Algorithm 1 maintains the stability and passivity of the
original system (1).
Proof of
Theorem 4. Equations
(25) and (26), when represented as matrix blocks, lead to equations (29) and
(30).
|
(29) |
|
(30) |
where:
|
(31) |
|
(32) |