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Modeling of discrete-time fractional-order state space systems using the balanced truncation method

JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 2017年第7期354卷 3008-3020页

作者： Stanislawski, Rafal Rydel, Marek Latawiec, Krzysztof J. Opole Univ Technol Dept Elect Control & Comp Engn Ul Proszkowska 76 PL-45758 Opole Opole Poland

This paper presents a new approach to approximation of linear time-invariant (LTI) discrete-time fractional-order state space SISO systems by means of the SVD-originated balanced truncation (BT) method applied to an FIR-based representation of the fractional-order system. This specific representation of the system enables to introduce simple, analytical formulas for determination of the Cholesky factor-izations of the controllability and observability Gramians, which contributes to significant improvement of the computational efficiency of the BT method. As a model reduction result for the fractional-order systems we obtain a low-order rational (integer-order) state space system. Simulation experiments show a high efficiency of the introduced methodology both in terms of the approximation accuracy of the model and low time complexity of the approximation algorithm. (C) 2017 The Franldin Institute. Published by Elsevier Ltd. All rights reserved.

关键词： Fractional-order system Discrete-time system Model order reduction balanced truncation method

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A New Scheme for the Approximation of Linear Dynamical Systems and Its Application to Controller Design

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CIRCUITS SYSTEMS AND SIGNAL PROCESSING 2024年第2期43卷 766-794页

作者： Suman, Santosh Kumar Maharana Pratap Polytech Dept Elect Engn Gorakhpur UP India

A new hybrid scheme of model order reduction (MOR) technique is developed for the simplification and control design of linear large-scale dynamical systems. MOR deals with a high-dimensional problem in a compact representation and plays an important role in fields where complicated simulation studies are still problematic in terms of simulation time. This paper proposes a strategy to derive a reduced order model (ROM) using balanced Singular Perturbation Approximation. This way reduces the higher order system into a reduced order model. The proposed method guarantees the preservation of all essential parameters of the original model in the reduced system. The results of the proposed method are compared with the results of the reduced order model by previously published work. Further, this ROM has been used for the design of a compensator for the large-scale original plant by using the proposed method. The compensator obtained by using the reduced model gives the approximately same time domain specification as the compensator obtained by using a large-scale original system, and the design of the compensator by using the reduced model is comparatively easier. The performance of the higher and lower-order controllers is also analysed with numerical examples in terms of time response characteristics and time domain specifications and performance indices.

关键词： Model order reduction Linear large-scale dynamical system Steady state value balanced truncation method Singular perturbation approximation Compensator Control design center dot

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Model order reduction of boiler system using nature-inspired metaheuristic optimization of PID controller

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DISCOVER APPLIED SCIENCES 2025年第5期7卷 1-36页

作者： Singh, Anurag Yadav, Shekhar Tiwari, Nitesh Nishad, Dinesh Kumar Khalid, Saifullah Madan Mohan Malaviya Univ Technol Dept Elect Engn Gorakhpur Uttar Pradesh India Dr Shakuntala Misra Natl Rehabil Univ Dept Elect Engn Lucknow Uttar Pradesh India IBM Multi Act Co Ltd Khartoum Sudan

Boiler system control presents significant challenges due to its complex, high-order dynamics, which make real-time control computationally demanding. Traditional model order reduction (MOR) techniques often compromise system accuracy, while conventional Proportional-Integral-Derivative (PID) tuning methods struggle with nonlinearities and dynamic uncertainties. This study proposes a dual-stage optimization framework that integrates balanced truncation-based model order reduction with nature-inspired metaheuristic algorithms for PID controller tuning to address these issues. The PID controllers are optimized using both classical methods such as Ziegler-Nichols (ZN), Simple Internal Model Control (SIMC), Approximate M-Constrained Integral Gain Optimization (AMIGO), and Chien-Hrones-Reswick (CHR), as well as advanced optimization techniques like Particle Swarm Optimization (PSO), Krill Herd Optimization (KHO), Harris Hawks Optimization (HHO), Moth-Flame Optimization (MFO), and Sparrow Search Optimization (SSO). Experimental results demonstrate that the PSO-optimized PID controller achieves a 20% reduction in settling time and a 14.68% improvement in Integral Square Error (ISE) compared to conventional tuning methods. The HHO-based approach improves overall performance by 15%, while SSO significantly reduces computational complexity by 65% while maintaining 98% system accuracy. Statistical analysis (p < 0.05) confirms the robustness of the proposed methodology, showing a 45% reduction in standard deviation compared to traditional approaches. The proposed framework offers a scalable, computationally efficient, and high-performance solution for industrial boiler control systems, ensuring improved stability, faster response times, and real-time adaptability over existing strategies.

关键词： Boiler system Soft computing Model order reduction balanced truncation method Nature-inspired computing Model predictive control System identification Industrial process control Computational intelligence Control system optimization balanced truncation

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Approximation of Fractional-Order Systems Using balanced truncation with Assured Steady-State Gain

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CIRCUITS SYSTEMS AND SIGNAL PROCESSING 2023年第10期42卷 5893-5923页

作者： Duddeti, Bala Bhaskar NIT Rourkela Elect Dept Rourkela Odisha India

This research presents a novel and stable approach for reducing the order of fractional-order systems using the advantage of a balanced truncation method and gain adjustment factor. The method can be applied to both commensurate and non-commensurate fractional systems. First, the fractional-order system is turned into an integer-order system using the Oustaloup approximation for non-commensurate systems and a simple mathematical replacement for commensurate systems. Then, balanced truncation is used to create a lower-order model. A gain factor is added to the reduced model to improve its accuracy in the steady state without changing how it works in the dynamic condition. This proposed method differs from other balanced truncation-based reduction methods that enhance steady-state response by using time moments, Pade equations, and Routh tables. These extra steps are optional with this method. The proposed strategy makes the step and Bode responses of the reduced systems more like Oustaloup's approximate higher-order model. The method's effectiveness is evaluated using various performance metrics such as integral square error, root mean square error, and H-infinity norm, as well as time response characteristics. Finally, inverse substitution converts the reduced integer-order model back to a commensurate fractional system. All simulations for the examples discussed in the study were conducted in MATLAB.

关键词： Model order reduction Fractional-order systems balanced truncation method Gain adjustment factor Oustaloup approximation

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Order Reduction of LTI Systems Using balanced truncation and Particle Swarm Optimization Algorithm

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CIRCUITS SYSTEMS AND SIGNAL PROCESSING 2023年第8期42卷 4506-4552页

作者： Duddeti, Bala Bhaskar Naskar, Asim Kumar Subhashini, K. R. Natl Inst Technol Rourkela Elect Engn Dept Rourkela India

This paper presents a novel hybrid model reduction method to simplify large-scale continuous-time single-input-single-output and multi-input-multi-output dynamic systems using the advantages of the balanced truncation method and the particle swarm optimization (PSO) algorithm. The balanced truncation method obtains the reduced model denominator coefficients to ensure stability. The PSO algorithm minimizes the integral square error between the step responses of the original system and the reduced model as much as possible. It leads to the optimal numerator coefficients. The advantage of the proposed approach is that, for optimizing reduced model numerator coefficients, the search space boundaries of the PSO algorithm are not entirely random. They are selected using the balanced truncated reduced model numerator. So, the suggested method avoids two significant problems with evolutionary algorithms: the arbitrary choice of search space and the longer simulation time. Four power system models and four numerical examples from the literature are considered to assess the effectiveness of the proposed reduction method. The step responses and Bode diagrams for the higher-order system and its corresponding reduced models are displayed. For comparison, statistics are tabulated based on the rise time, the settling time, the peak overshoot, the integral square error, and the root-mean-square error. The proposed method ensures stability and other features of the higher-order system in the reduced model. The performance measure values and the time domain characteristics demonstrate the ability and effectiveness of the proposed approach. All the case studies use a MATLAB simulation environment.

关键词： Large-scale dynamic systems Model order reduction Particle swarm optimization Performance indices Soft computing technique balanced truncation method SISO and MIMO systems

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Study on key factors affecting the high-order building model order reduction for model predictive control application

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ENERGY AND BUILDINGS 2023年第1期296卷

作者： Chen, Qiong Li, Nan Wuhan Univ Technol Sch Civil Engn & Architecture Wuhan 430070 Peoples R China Chongqing Univ Natl Ctr Int Res Low carbon & Green Bldg Minist Sci & Technol Chongqing 400045 Peoples R China

The reduced-order model (ROM) can highly reduce computation costs while maintaining high high-fidelity performance for model predictive control's application by applying the model order reduction method. However, the physical meaning of the building thermal model order reduction at the theoretical level was not been fully explored in this numerical process, although the model order reduction process and the results were unquestionable at the mathematical level. The interpretability of the building low-dimensional model at the physical level is a necessary aspect to improve the reliability of the model. The unclear numerical relationships between the key influential factors of the building's thermal property and the ROM performance identified by accuracy and computation cost implied the potential risk of ROM mismatch under certain dynamic conditions. In this research, the comprehensive research on impacts of potential key variables including the outdoor and indoor thermal frequency, radiant heating system & all-air system, building envelopes' thermal property, discretization timesteps, and model outputs on ROM's performance were investigated through detailed parametric simulations. In practical engineering, the best-performing ROM can be selected based on the frequency and amplitude of thermal disturbance fluctuations to ensure the accuracy of the ROM while minimizing the computational cost. At least one order was required to identify the indoor heat load which meant that small indoor heat load cannot be neglected in the model order reduction process and larger heat loads required ROM with more orders in consideration of model accuracy. ROMs of all-air systems with less thermal inertia were more robust to the heating power changes than that of radiant heating systems with large thermal inertia and significant time lag. Building envelope materials with the same or approximate thermal properties can be simply characterized with fewer nodes. The greater the difference i

关键词： Model order reduction Building thermal model High-fidelity Model computation cost balanced truncation method

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A Modified balanced truncation method and its Application to Model Reduction of Power System

A Modified Balanced Truncation Method and its Application to...

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General Meeting of the IEEE-Power-and-Energy-Society (PES)

作者： Huang, Chongxin Zhang, Kaifeng Dai, Xianzhong Tang, Wenjie Southeast Univ Key Lab Measurement & Control CSE Minist Educ Sch Automat Nanjing 210096 Jiangsu Peoples R China

ISBN: (纸本)9781479913039

The balanced truncation method (BTM) can make the reduced model preserve the controllability, observability, stability, and dynamic response of the original model, but some steady-state deviations between the reduced model and the original model exist. On the premise of keeping the merits of the BTM, this paper modifies the BTM to narrow the deviations by introducing a gain factor into the transfer function matrix of the reduced model to adjust the steady-state value of the reduced model, but not to change the dynamic behavior. For verifying the modified BTM, we apply the method to the model reduction of a typical two-machine system. The testing results indicate that the modified method can decrease the deviations, lower the order of the system, and retain dynamic behaviors of the original model.

关键词： balanced truncation method dynamic equivalents model reduction power system

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Model reduction techniques for high-rise buildings and its reduced-order controller with an improved BT method

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STRUCTURAL ENGINEERING AND MECHANICS 2021年第3期78卷 305-317页

作者： Chen, Chao-Jun Teng, Jun Li, Zuo-Hua Wu, Qing-Gui Lin, Bei-Chun Harbin Inst Technol Sch Civil & Environm Engn Shenzhen Peoples R China Hong Kong Polytech Univ Dept Civil & Environm Engn Hong Kong Peoples R China

An AMD control system is usually built based on the original model of a target building. As a result, the fact leads a large calculation workload exists. Therefore, the orders of a structural model should be reduced appropriately. Among various model-reduction methods, a suitable reduced-order model is important to high-rise buildings. Meanwhile, a partial structural information is discarded directly in the model-reduction process, which leads to the accuracy reduction of its controller design. In this paper, an optimal technique is selected through comparing several common model-reduction methods. Then, considering the dynamic characteristics of a high-rise building, an improved balanced truncation (BT) method is proposed for establishing its reduced-order model. The abandoned structural information, including natural frequencies, damping ratios and modal information of the original model, is reconsidered. Based on the improved reduced-order model, a new reduced-order controller is designed by a regional pole-placement method. A high-rise building with an AMD system is regarded as an example, in which the energy distribution, the control effects and the control parameters are used as the indexes to analyze the performance of the improved reduced-order controller. To verify its effectiveness, the proposed methodology is also applied to a four-storey experimental frame. The results demonstrate that the new controller has a stable control performance and a relatively short calculation time, which provides good potential for structural vibration control of high-rise buildings.

关键词： flexible and high-rise buildings active mass damper/driver balanced truncation method model reduction

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Fast simulation and high-fidelity reduced-order model of the multi-zone radiant floor system for efficient application to model predictive control

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ENERGY AND BUILDINGS 2021年 248卷 111210-111210页

作者： Chen, Qiong Li, Nan Chongqing Univ Minist Sci & Technol Natl Ctr Int Res Low Carbon & Green Blds Chongqing 400045 Peoples R China

Model predictive control, as the most popular intelligent advanced control technology in recent years, is increasingly applied to building air conditioning systems to achieve adaptive and energy-efficient operation of the system, however, model predictive control imposes higher requirements on the computational efforts of the model. In this research, a complex coupled heat transfer model of the increasingly popular radiant floor system is analyzed and a full-order model of the radiant floor is developed. Then, after controllability and observability analysis, a fast high-fidelity reduced-order model that can accurately characterize the dynamic thermal performance of the original model is proposed applying the balanced truncation method. The accuracy of the reduced-order model was verified by comparing the radiant floor surface temperature, return water temperature, zone air temperature, and unheated average internal surface temperature with experimental test results, the TRNSYS model, and the full-order model. The computation time of the reduced-order model for open-loop simulation was reduced by 86% to 98% compared to the TRNSYS model. More importantly, the reduced-order model can effectively reduce the complexity of the model predictive control, resulting in a 38% to 78% saving in computation time compared to the full-order model, significantly improving computational efficiency and enhancing the robustness of the model predictive control. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Model order reduction balanced truncation method Radiant floor system Model predictive control Building simulation

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Spherical Bessel transform via exponential sum approximation of spherical Bessel function

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JOURNAL OF COMPUTATIONAL PHYSICS 2018年 355卷 426-435页

作者： Ikeno, Hidekazu Osaka Prefecture Univ Org Res Promot Res Ctr 21 Century NanoSq Res InstNaka Ku 1-2 Gakuen Cho Sakai Osaka 5998570 Japan JST PRESTO 4-1-8 Hon Cho Kawaguchi Saitama 3320012 Japan

A new algorithm for numerical evaluation of spherical Bessel transform is proposed in this paper. In this method, the spherical Bessel function is approximately represented as an exponential sum with complex parameters. This is obtained by expressing an integral representation of spherical Bessel function in complex plane, and discretizing contour integrals along steepest descent paths and a contour path parallel to real axis using numerical quadrature rule with the double-exponential transformation. The number of terms in the expression is reduced using the modified balanced truncation method. The residual part of integrand is also expanded by exponential functions using Prony-like method. The spherical Bessel transform can be evaluated analytically on arbitrary points in half-open interval. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Hankel transform Spherical Bessel transform Bessel function Oscillatory integrals balanced truncation method

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