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) |
|
(33) |
Following this, the reduced-order system
satisfies the Lyapunov condition expressed in equation (34).
|
(34) |
and the Riccati equation (35).
|
(35) |
xB1 is a positive definite, symmetric, and
diagonal matrix, meeting the requirements of Remark 1, Theorem 1, and Theorem
2, so the reduced-order system obtained from Algorithm 1 preserves both the
stable and passive of the original system (1).
Corollary 2. The system was reduced using Algorithm 1, which employs the
controllability Gramian xBc and the observability Gramian xBo. Both Gramians exhibit diagonal, symmetric, and positive definite
properties, containing r Hankel singular values derived from the
original mixed-balanced system.
Proof of Corollary 2. From equations (29) to (32), it follows that what must be proven.
Theorem 5. Considering system (1) satisfying Remark 1. When applying the MBT
algorithm for order reduction, the upper bound of error is defined by condition (36).
|
(36) |
Proof of Theorem 5. Transforming equation (15) into equation (37)
|
(37) |
where is as in equation (38)
|
(38) |
System
(1) is a mixed-balanced system, and equations (2) and (5) are consequently
converted into equations (39) and (40).
|
(39) |
|
(40) |
These equations are two Lyapunov equations. Based on the transformations and demonstrations in the BT algorithm, the error according to the Hinf norm between the original and reduced-order systems satisfies Theorem 5.
Illustrative example
Considering
the RLC network as a model of a transmission line (Akram et
al., 2020), and selecting the number of
nodes k = 8, the order of the system is n = 15. By convention,
the state variables represent the voltage across Ci,
denotes
the current through Lj, u is the input voltage, and y
is the output current, where i ranges from 1 to 2k and j
ranges from 1 to 2k-1.
We apply
the BT, PRBT, and MBT algorithms to reduce the order of the RLC ladder network
model from r = 1 to r = n-1. Figure 1 illustrates the Absolute
Error plot using the H-infinity norm between the reduced-order and original
systems. Table 1 shows the absolute errors corresponding to each order r of the
model.
Figure 1 Absolute error plot with decreasing order of r
The plot in Figure 1 depicts the Absolute Hinf Error versus the model order r for three model reduction algorithms: BT, PRBT, and
MBT. From this result, we have the following Analysis and Observations:
-
MBT algorithm: MBT shows
stable and low error values across all model orders. This indicates that MBT is
highly reliable and maintains the accuracy of the reduction order model. The
error curve for MBT (green dash-dot line) suggests that MBT is an effective
method for model reduction.
-
BT algorithm: BT's error
curve (red dashed line) is stable and does not show significant variations,
indicating that low errors are maintained across different model orders. BT,
like MBT, proves to be a reliable method for model reduction.
-
PRBT algorithm: PRBT shows
significant fluctuations in error values across different model orders. As the
order decreases, the error of PRBT (represented by the blue solid line)
increases rapidly, indicating a lack of stability. The error values vary
considerably, depending on the chosen model order. This suggests that PRBT
might be less reliable and sensitive to the choice of r.
From Table 1, in conjunction with the numerical results, several insights
can be gleaned:
-
Both MBT and BT algorithms
show consistent and stable error reduction across all model orders. They
progressively reduce the H-infinity norm error without significant
fluctuations, making them both reliable choices for model order reduction.
-
While PRBT exhibits large
fluctuations in errors, MBT maintains a steady, demonstrating superior
stability
The Hankel singular Values (HSV)
of the transmission line model, upon transformation into a mixed-balanced
system, are detailed in Table 2. In table 2, the HSV gradually decreases as the
order r increases. This trend aligns with both theoretical expectations
and practical observations, as smaller HSV values indicate a lesser loss of
information from the original system, consequently resulting in a proportional
increase in the reduction error as the system order decreases.
By comparing Table 2 with the
error reduction upper bound formula specified by the MBT method in equation
(36), the maximum values of the estimated error are listed in the
"Proposed error" column of Table 1. When these predicted errors are
compared with the real errors shown in the "MBT error" column of
Table 1, it is clear that the formula presented in Theorem 6 is accurate.
Performing order reduction on
the RLC ladder network model to achieve a 3rd-order representation, we generate
Absolute Error in Amplitude (dB) vs. Frequency (rad/sec), Phase Error (dec) vs.
Frequency (rad/sec), and Absolute Error (Amplitude) vs. Time (second) plots for
the three methods in Figure 3 and Figure 4, respectively.
Table 1 Comparison of H-infinity Norm Errors between BT, PRBT, and MBT.
Model Order (r) |
BT Error |
PRBT Error |
MBT Error |
Proposed error |
1 |
1.749991 |
1.866309 |
1.717820 |
13.65631 |
2 |
1.734798 |
4.938759 |
1.574066 |
12.31516 |
3 |
1.739218 |
8.155481 |
1.637961 |
10.98345 |
4 |
1.646226 |
2.999828 |
1.525381 |
9.707632 |
5 |
1.534106 |
3.504173 |
1.599395 |
8.461003 |
6 |
1.502363 |
7.664413 |
1.353905 |
7.287345 |
7 |
1.364971 |
35.10419 |
1.451378 |
6.160561 |
8 |
1.328143 |
161.3269 |
1.299514 |
5.111082 |
9 |
1.240640 |
4610.024 |
1.220345 |
4.118088 |
10 |
1.154465 |
2971.094 |
1.127298 |
3.196535 |
11 |
1.083673 |
165.6717 |
0.985305 |
2.329888 |
12 |
1.021401 |
57.23219 |
0.917855 |
1.518259 |
13 |
0.9848805 |
60.56017 |
0.901017 |
0.945749 |
14 |
0 |
0 |
0 |
0 |
Table 2 The HSV of
mixed-balanced system.
?i |
HSV |
?i |
HSV |
?i |
HSV |
1 |
1.226872736948283 |
6 |
0.586829045407931 |
11 |
0.433323477959059 |
2 |
0.670576895359906 |
7 |
0.563391797949794 |
12 |
0.405814550194131 |
3 |
0.665852832023017 |
8 |
0.524739486454494 |
13 |
0.386254508104516 |
4 |
0.637910984013295 |
9 |
0.496497172492625 |
14 |
0.372874846777396 |
5 |
0.623314517294628 |
10 |
0.460776660475757 |
15 |
0 |
From the Frequency Domain Error
plot in Figure 3, we have the Comparison (Magnitude and Phase Errors) as
follows:
- Magnitude Error:
+ error_BT (red dashed line):
The magnitude error increases from approximately -20 dB at low frequencies to 0
dB at high frequencies, suggesting that the BT algorithm exhibits higher error
compared to both PRBT and MBT.
+ error_PRBT (blue solid line): The magnitude error of PRBT is lower and
more stable than BT, but shows slight fluctuations in the mid-frequency range.
+ error_MBT` (green short dashed line): The magnitude error of MBT
closely matches PRBT at high and low frequencies, indicating that MBT performs
as well or better than PRBT and significantly better than BT.
- Phase Error:
+ error_BT: The phase error for
BT is relatively low and stable at low and mid frequencies but increases
slightly at high frequencies.
+ error_PRBT: The phase error
for PRBT is higher than BT, especially at low and mid frequencies.
+ error_MBT: The phase error for
MBT is lower than PRBT and comparable to BT at high frequencies, indicating
that MBT maintains better phase accuracy.
- Accuracy in Frequency Domain:
+ BT exhibits larger errors in
both magnitude and phase within the frequency domain, in comparison to PRBT and
MBT.
+ PRBT improves magnitude
accuracy but has higher phase errors.
+ MBT maintains the smallest
errors in both magnitude and phase, especially in the high-frequency range,
indicating better accuracy in the frequency domain.
Figure 2 Frequency
Domain Error plot between BT, PRBT, and MBT Algorithms
Figure 3 Time Domain Error plot between BT, PRBT, and MBT Algorithms
From The Time Domain Error plot
as Figure 4, we have the Comparison between BT, PRBT, and MBT algorithms as
follows:
- error_BT: The time domain
error for BT is smaller and more stable initially but tends to increase over
time.
- error_PRBT: The time domain
error for PRBT oscillates around zero, but with larger oscillations than BT.
- error_MBT: The time domain
error for MBT is very small and follows PRBT closely, indicating performance
that is equivalent to or somewhat better than PRBT but larger than BT.
Overall evaluation:
- MBT proves to be a performing
method in both the frequency and time domains, with small and stable errors.
- PRBT is also a good method,
particularly for reducing magnitude errors but has higher phase errors.
- BT performs worse compared to
PRBT and MBT, with larger errors in the frequency domain.
- Both the MBT and BT algorithms
demonstrate high reliability for model order reduction, offering stable and
consistent performance across various orders. They both achieve high accuracy
at higher orders, making them suitable for practical applications requiring
precise reduced models.
Therefore, MBT is the preferred
method for minimizing errors and maintaining the highest accuracy in model
order reduction systems. MBT proves to be a superior choice when compared to
PRBT due to its consistent, low error performance and preserved passivity,
while it matches the robustness, reliability, and stability of BT.
5.1. Contributions
to Scientific Theory of the Research Results
The research team developed a model
order reduction algorithm (Algorithm 1: Reduce Model Order Using Mixed Balanced
Truncation, MBT) to address the
limitations of two original algorithms. Specifically, MBT outperforms BT by
preserving both stability and passivity, offering lower computational costs and
reducing errors compared to PRBT.
During the development of MBT, the authors presented mathematical
arguments accompanied by proofs, including Definition 1: Mixed-balanced system
definition; Theorem 3: Existence of non-singular transformation; Lemma 1:
Eigenvalue invariance under transformation; Lemma 2: Achieving mixed-balance
via transformation; Corollary 1: Properties of the mixed-balanced system; Theorem 4: Stability
and passivity of reduced-order systems; Corollary 2: Gramian properties in
reduced systems; Theorem 5: Error bound in MBT algorithm.
The MBT reduction algorithm simplifies passive circuits with numerous
state variables, minimizing computational costs and optimizing the simulation
and analysis of high-order systems.
The algorithm and theoretical findings can be applied to the design,
development, testing, evaluation, measurement support, response prediction,
risk warning, and functional verification of high-order electrical systems
using lower-order circuit models.
The content and results of this research can serve as reference material
for learning, research, and teaching on system identification, model order
reduction, and circuit design and simulation.
The study provides knowledge and source codes to support the development
of a reduction toolbox for linear systems in MATLAB.
In resource-constrained environments:
- Complex systems with large, multi-source datasets: Obtaining
comprehensive input signals poses challenges in designing, analyzing,
surveying, evaluating, predicting, modeling, identifying, and simulating
systems. MBT focuses on the most impactful input and output signals identified
through preliminary assessments of measurable operational parameters. It does
not require a full system model. Instead, it identifies key signals, eliminates
less impactful components, and generates a reduced-order model that effectively
simulates critical responses without detailed data. This method optimizes
resource usage, ensures system performance, and preserves key feedback
properties.
- Based on statistical datasets: MBT remains crucial in reducing
complexity by focusing on the most significant state variables. This not only simplifies the system but also
accelerates signal processing and reduces computational load, ensuring enhanced
efficiency while preserving the core dynamic characteristics of the system.
5.2. Computational
Costs of the Algorithms
The BT, PRBT, and MBT algorithms
rely on the principle of Gramian balancing, followed by truncation of balanced
equivalent system matrices. The computational complexity differences stem from
solving matrix equations to determine the observability and controllability
Gramians of the original system.
Lyapunov equation complexity: . If
, the
term dominates. If
, the
term dominates. For
, the overall complexity is
.
Performance
Consideration: The complexity is primarily determined by the sizes of n
and m. In cases where m is relatively small, the computational
cost is dominated by the matrix-matrix multiplications involving A, P, and E. Solving two Lyapunov equations in BT incurs
this cost twice.
Riccati equation complexity: . If
, the complexity is dominated by
. If
, the complexity is
dominated by
. For
, the overall complexity is
. The inversion
of
and the multiplication involving B, C and D contribute significantly to the complexity
when mmm is large. This is critical for performance optimization. Solving two
Riccati equations in PRBT doubles this cost.
In MBT, determining the Gramians involves solving one Lyapunov equation
and one Riccati equation, making MBT less computationally expensive than PRBT.
5.3. Limitations
of the MBT Algorithm
While MBT achieves better
computational efficiency and lower reduction errors than PRBT, its complexity
remains higher than BT, with greater deviations from the original model.
As with most algorithms, MBT's
accuracy and processing speed depend on factors such as software data types,
matrix solver precision, hardware configuration, firmware performance,
programming language, implementation optimization, and the complexity of the
original system's data.
MBT is designed for linear
systems (1) meeting requirements in Remark 1. Systems not satisfying Remark 1
require intermediate transformations to conform to the required format.
Solving complex matrix equations
in MBT can lead to increased computational costs for systems with large
datasets, posing challenges for hardware with limited processing capabilities.
5.4. Development
Directions
Reduce algorithm complexity by
employing Newton iteration, low-rank approximations, or rational Krylov
subspace methods.
To minimize reduction-induced
errors, integrate optimization techniques (De Guzman et al., 2024; Jusuf et al.
2024; Wichapa et al. 2024; Nitnara and Tragangoon, 2023; Hendrarini et
al., 2022), with objectives such as error
minimization and preservation of system physical properties.
For nonlinear systems, preprocess
using linearization algorithms before applying MBT.
For unstable systems or those
with mixed stable/unstable components:
+ Decompose into stable and
unstable subsystems, apply MBT to the stable component, and combine the
reduced-order model with the unstable component.
+ Use partial stabilization
techniques to transform the unstable system into a stable one before applying
MBT.
In this paper, we introduced a
novel Mixed Balanced Truncation (MBT) algorithm tailored for reducing linear
time-invariant continuous-time descriptor systems, specifically within the
context of electrical and electronic circuit simulations. Our approach aimed to
amalgamate the benefits of Balanced Truncation (BT) and Positive-real balanced
truncation (PRBT) while mitigating specific drawbacks. The MBT algorithm
demonstrated superior performance in consistently maintaining low error values
across various model orders, indicating high reliability for model order
reduction with minimal loss of accuracy. Compared to BT and PRBT, MBT showed
marked improvement in error metrics, with a steady and predictable decrease in
errors, significantly outperforming PRBT, which had large fluctuations and
sensitivity to reduction order. MBT retained the essential properties of the
original system, including stability and passivity, as confirmed by theoretical
proofs and numerical simulations. Its application to an RLC ladder network
model effectively reduced computational complexity while preserving the
original system's dynamic behavior, which is valuable for resource-limited
environments. Additionally, our study contributes to the theoretical
understanding of model order reduction by providing new insights into the
transformation and balancing of descriptor systems supported by established
theorems, lemmas, and corollaries. Future work could explore further
optimizations and extensions of the MBT approach to other types of systems and
applications, thereby broadening its impact and utility within electronics and
electrical engineering.
This research was financially supported by the Program of the Ministry of Education and Training of Vietnam under grant number B2023-TNA-17.
Author Contributions
Conceptualization, Methodology, Funding acquisition- Huy-Du Dao(H.-D D); Investigation, Data curation, Validation Ngoc-Kien Vu(N.-K V);Investigation, Validation Van-Ta Hoang (V.-T H); Formal analysis, Software, Resources, Writing—original draft, Writing—review & editing, Thanh-Tung Nguyen (T.-T N); Formal analysis, Supervision, Project administration Hong-Quang Nguyen (H.-Q N). All authors have read and agreed to the published version of the manuscript.
Conflict of Interest
The
authors declare no conflicts of interest.
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