Conventional and Evolutionary Order Reduction Techniques for Complex Systems

2021 ◽  
Vol 16 (4) ◽  
pp. 74-98
Author(s):  
Abha Kumari ◽  
C. B. Vishwakarma
Keyword(s):  
Large Scale ◽  
Order Reduction ◽  
Original System ◽  
Conventional Technique ◽  
Stability Equation ◽  
Padé Approximations ◽  
Time Frequency ◽  
Reduction Techniques ◽  
Pade Approximations ◽  
The Stability

Order reduction of the large-scale linear dynamic systems (LSLDSs) using stability equation technique mixed with the conventional and evolutionary techniques is presented in the paper. The reduced system (RS) is obtained by mixing the advantages of the two methods. For the conventional technique, the numerator of the RS is achieved by using the Pade approximations and improved Pade approximations, whereas the denominator is obtained by the stability equation technique (SET). For the evolutionary technique, numerator of the RS is obtained by minimizing the integral square error (ISE) between transient responses of the original and the RS using the genetic algorithm (GA), and the denominator is obtained by the stability equation method. The proposed RS retains almost all the essential properties of the original system (OS). The viability of the proposed technique is proved by comparing its time, frequency responses, time domain specifications, and ISE with the new and popular methods available in the literature.

Application of projection-based model reduction to finite-element plate models for two-dimensional traveling waves

2016 ◽  
Vol 28 (14) ◽  
pp. 1886-1904 ◽  
Author(s):  
Vijaya VN Sriram Malladi ◽  
Mohammad I Albakri ◽  
Serkan Gugercin ◽  
Pablo A Tarazaga
Keyword(s):  
Finite Element ◽  
Model Reduction ◽  
Traveling Waves ◽  
Large Scale ◽  
Fe Model ◽  
Plate Model ◽  
Reduction Techniques ◽  
State Frequency ◽  
Scale Down ◽  
The Stability

A finite element (FE) model simulates an unconstrained aluminum thin plate to which four macro-fiber composites are bonded. This plate model is experimentally validated for single and multiple inputs. While a single input excitation results in the frequency response functions and operational deflection shapes, two input excitations under prescribed conditions result in tailored traveling waves. The emphasis of this article is the application of projection-based model reduction techniques to scale-down the large-scale FE plate model. Four model reduction techniques are applied and their performances are studied. This article also discusses the stability issues associated with the rigid-body modes. Furthermore, the reduced-order models are utilized to simulate the steady-state frequency and time response of the plate. The results are in agreement with the experimental and the full-scale FE model results.

scholarly journals Approximation of stability radii for large-scale dissipative Hamiltonian systems

2020 ◽  
Vol 46 (1) ◽  
Author(s):  
Nicat Aliyev ◽  
Volker Mehrmann ◽  
Emre Mengi
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Order Reduction ◽  
Interpolation Property ◽  
Element Model ◽  
Positive Semidefinite ◽  
Stability Radius ◽  
Linear Time Invariant ◽  
Reduction Techniques ◽  
Stability Radii

Abstract A linear time-invariant dissipative Hamiltonian (DH) system $\dot x = (J-R)Q x$ẋ=(J−R)Qx, with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625–1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + BΔCH for given matrices B, C, and another with respect to Hermitian perturbations in the form R + BΔBH,Δ = ΔH. We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake.

Extended approaches based on Genetic Algorithms (GAs) for MIMO singular systems approximation

2016 ◽  
Vol 40 (1) ◽  
pp. 148-154
Author(s):  
Amel BH Adamou-Mitiche ◽  
Lahcène Mitiche
Keyword(s):  
Genetic Algorithms ◽  
Singular Systems ◽  
Singular System ◽  
High Dimensionality ◽  
Reduced Order Models ◽  
Stability Equation ◽  
Reduced Order ◽  
Reduction Techniques ◽  
The Stability

In this paper we present an extension of three important model reduction techniques: namely, the stability equation, the modified pole clustering and the dominant modes methods for conventional (regular) systems to reduce complexity relating to high dimensionality of mathematical models representing physical, generalized (also called singular) systems. Combining these methods to Genetic Algorithms’ tools and exploiting a special representation base where a full order singular system is deflating into proper and improper subsystems, different natures of stable, optimal low order models are obtained. To show the effectiveness of the proposed algorithms, a numerical example is given, where six approximants are derived from a multi-input multi-output singular system. By the use of two optimal norms, the MOR errors are quantified and permits to conclude to the quality of the proposed reduced order models.

scholarly journals Reduction of Large-Scale Dynamical Systems by Extended Balanced Singular Perturbation Approximation

2020 ◽  
Vol 5 (5) ◽  
pp. 939-956 ◽  
Author(s):  
Santosh Kumar Suman ◽  
Awadhesh Kumar
Keyword(s):  
Singular Perturbation ◽  
Large Scale ◽  
Linear Time ◽  
Order Reduction ◽  
Original System ◽  
Simplified Approach ◽  
Linear Time Invariant ◽  
Time Invariant ◽  
Time Systems ◽  
Perturbation Approximation

A simplified approach for model order reduction (MOR) idea is planned for better understanding and explanation of large- scale linear dynamical (LSLD) system. Such approaches are designed to well understand the description of the LSLD system based upon the Balanced Singular Perturbation Approximation (BSPA) approach. BSPA is tested for minimum / non-minimal and continuous/discrete-time systems valid for linear time-invariant (LTI) systems. The reduced-order model (ROM) is designed to preserved complete parameters with reasonable accuracy employing MOR. The Proposed approach is based upon retaining the dominant modes (may desirable states) of the system and eliminating comparatively the less significant eigenvalues. As the ROM has been derived from retaining the dominant modes of the large- scale linear dynamical stable system, which preserves stability. The strong aspect of the balanced truncation (BT) method is that the steady-state values of the ROM do not match with the original system (OS). The singular perturbation approximation approach (SPA) has been used to remove this drawback. The BSPA has been efficaciously applied on a large-scale system and the outcomes obtained show the efficacy of the approach. The time and frequency response of an approximated system has been also demonstrated by the proposed approach, which proves to be an excellent match as compared to the response obtained by other methods in the literature review with the original system.

Suboptimal Bilinear Routh Approximant for Discrete Systems

2004 ◽  
Vol 128 (3) ◽  
pp. 742-745 ◽  
Author(s):  
Younseok Choo
Keyword(s):  
Impulse Response ◽  
Approximation Method ◽  
Order Reduction ◽  
Original System ◽  
Discrete Systems ◽  
Reduced Model ◽  
The Stability

Recently an improved bilinear Routh approximation method has been suggested for the order reduction of discrete systems. In the method, the last α and β parameters of a reduced model were replaced by new parameters so that the impulse response energy of an original system is also preserved in the reduced model without destroying the stability preserving and time-moments matching properties. In this paper a new and simple improvement is proposed from which one can find a suboptimal bilinear Routh approximant. Compared to the previous result, the approach of this paper has an advantage that the improvement is always guaranteed.

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