Model Order Reduction Of Systems Research Articles

A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine a classical finite volume element (FVE) method with POD method to establish a reduced FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional viscoelastic problem with real practical applied background, and analyze the errors between the reduced POD FVE solution and the classical FVE solution so as to provide scientific theoretic basis for service applications. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced FVE formulation based on POD method is feasible and efficient for solving two-dimensional viscoelastic problem.

Effects of strong resonance in tuning of multiple power system stabilisers

The phenomena of strong resonance occur due to dynamic modal interaction between the power system's modes when one/two system or controller parameters are varied. This study presents the effect of strong resonance while designing a single or multiple power system stabilisers (PSS) in multi-machine power systems that can lead to instability of the system. The effect of strong resonance is investigated in test and practical power systems. It is studied with full-order and reduced-order system models based on the method of perturbations suggested by Seiranyan. The authors also suggest a mechanism to mitigate strong resonance while tuning one or more PSS. The test systems considered are three-machine, ten-bus system and a four-machine, 11-bus practical system, a part of the southern-grid Indian power system.

Model Order Reduction for Pre-Stressed Harmonic Analysis of Micromechanical Beam Resonators

A second-order system model order reduction method for pre-stressed harmonic analysis of electrostatically actuated microbeams is demonstrated, which produces a low dimensional approximation of the original system and enables a substantial reduction of simulation time. The moment matching property for second-order dynamic systems is studied and the block Arnoldi algorithm is adopted for the generation of the Krylov subspace, which extracts the low order model from the discretized system assembled through finite element analysis. The difference between two successive reduced models suggests the choice of the order for the reduce model. A detailed comparison research among the full model and the reduced models is performed. The research results confirm the effectiveness of the presented method.

Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems

AbstractData-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework.

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter ν = 10 −3, the two-dimensional flow past a cylinder at Reynolds number Re = 200, and the three-dimensional flow past a cylinder at Reynolds number Re = 1000.

Control of nonlinear systems represented by Galerkin models using adaptation-based linear parameter-varying models

This paper studies the control of nonlinear Galerkin systems, which are an important class of nonlinear systems that arise in reduced-order modeling of infinite-dimensional systems. A novel approach is proposed in which a linear parameter-varying (LPV) model representing the Galerkin model is built, where the parameter variation is dictated by a specially designed adaptation scheme. The controller design is then carried out on the simpler LPV model, instead of dealing directly with the complicated nonlinear Galerkin system. An automatically scheduled H-infinity controller is designed using the LPV model, and it is proven that this controller will indeed achieve the desired stabilization when applied to the nonlinear Galerkin model. The approach is illustrated with an example on cavity flow control, where the design is seen to produce satisfactory results in suppressing unwanted oscillations.

Control of an oil-heated, fractal-like branching microchannel desorber

A PID controller was designed to improve the performance of an oil-heated, fractal-like branching microchannel desorber for use in an ammonia-absorption refrigeration system. System identification techniques were employed to develop single-input, single-output reduced order models relating the oil mass flow rate and the rectified circulation ratio ( f * ) for four different operating conditions. PID controllers were designed for each of these reduced order system models and the closed-loop response of the desorber was simulated for various operating conditions and controller combinations. The preliminary controller designs were refined by examining controlled desorber performance in an experimental flow loop. The resulting PID controller yielded a closed-loop response with a faster rise time and lower overshoot than the simulated controllers. The tuned controller regulated the desired rectified circulation ratio under the operating conditions studied, controlling both the amount and concentration of the ammonia refrigerant generated. Prescribed variations in f * were tracked by the controlled system with a settling time of approximately 10 min. Controlled performance was robust to external disturbances in the strong solution flow rate and the oil temperature, regulating f * about the desired value with a settling time similar to that observed during the tracking tests.

Considerations on a Mass-Based System Representation of a Pneumatic Cylinder

Pneumatic actuators can be advantageous over electromagnetic and hydraulic actuators in many servo motion applications. The difficulty in their practical use comes from the highly nonlinear dynamics of the actuator and control valve. Previous works have used the cylinder’s position, velocity, and internal pressure as state variables in system models. This paper replaces pressure in the state model with the mass of gas in each chamber of the cylinder, giving a better representation of the system dynamics. Under certain circumstances, the total mass of gas in the cylinder may be assumed to be constant. This allows development of a reduced-order system model.

LTV Model Reduction With Upper Error Bounds

A new approach for computing upper error bounds for reduced-order models of linear time-varying systems is presented. It is based on a transformation technique of the Hankel singular values using positive-real, odd incremented functions. By applying such time-varying functions, the singular values to be removed can be forced to become equal and constant, so that they can be reduced. Two variations of this method are proposed: one for finite-time horizons and the other for infinite-time problems including periodic systems.

Missing Point Estimation in Models Described by Proper Orthogonal Decomposition

This paper presents a new method of missing point estimation (MPE) to derive efficient reduced-order models for large-scale parameter-varying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projection-based model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of data-dependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25% of the spatial grid points without compromising the accuracy of the reduced model.

Gramian-Based Reduction Method Applied to Large Sparse Power System Descriptor Models

This paper presents an efficient linear system reduction method that computes approximations to the controllability and observability gramians of large sparse power system descriptor models. The method exploits the fact that a Lyapunov equation solution can be decomposed into low-rank Choleski factors, which are determined by the alternating direction implicit (ADI) method. Advantages of the method are that it operates on the sparse descriptor matrices and does not require the computation of spectral projections onto deflating subspaces of finite eigenvalues, which are needed by other techniques applied to descriptor models. The gramians, which are never explicitly formed, are used to compute reduced-order state-space models for large-scale systems. Numerical results for small-signal stability power system descriptor models show that the new method is more efficient than other existing approaches.

System reduction using eigen spectrum analysis and Padé approximation technique

A mixed method is proposed for deriving reduced-order models of high-order linear time invariant systems using the combined advantages of eigen spectrum analysis and the Padé approximation technique. The denominator of the reduced-order model is found by eigen spectrum analysis, the dynamics of the numerator are chosen using the Padé approximation technique. This method guarantees stability of the reduced model if the original high-order system is stable. The method is illustrated by three numerical examples.

확률론적 발란스 방법을 이용한 제어용 모델의 축소

Recently, dynamic system has been enlarged and is normally exposed to various types of disturbance. Thus designing controller for these dynamic systems under random disturbance is not practically easy. As a result, the exact analysis for the system which is exposed to various irregular disturbance is quite important. In order to perform analysis, conventional BMR(balance model reduction) method is adopted and applied to moment equation in stochastic domain. Reliable reduced order system model has been obtained.

Karhunen–Loeve analysis and order reduction of the transient dynamics of linear coupled oscillators with strongly nonlinear end attachments

The Karhunen–Loeve (K–L) decomposition method has become a popular technique to create low-dimensional, reduced-order models of dynamical systems. In this paper this technique is applied to a multi-degree-of-freedom chain of linear coupled oscillators with a strongly nonlinear (nonlinearizable), lightweight end attachment. By performing K–L decomposition we show that the lightweight nonlinear attachment (possessing 0.5% of the total mass of the chain) can affect the global dynamics of the linear chain, exhibiting nonlinear energy-pumping phenomena; that is, irreversible passive targeted energy transfers from the linear chain to the nonlinear end attachment, where this energy is locally confined and dissipated without ‘spreading back’ to the primary system. It is shown that the occurrence of energy pumping can be identified by studying the dominant K–L modes of the dynamics, as well as, the energy distribution among them. Moreover, by comparing the action of the strongly nonlinear attachment to the classical linear vibration absorber, we show robustness of passive nonlinear energy absorption over wide parameter ranges. On the other hand, the case-sensitive nature of K–L-based reduced-order models has always been a constraint for K–L decomposition, since one cannot quantify a priori the error bound of such low-dimensional reduced-order models when different initial conditions are applied to the system. To alleviate this constraint, the paper proposes a multiple correlation coefficient (MCC) as a quantitative measure to effectively assess the applicability of a K–L-based reduced-order model derived for a specific set of initial conditions to a small neighborhood of initial conditions containing that initial state. The derived reduced-order models are validated through reconstruction of the system responses and comparisons to direct numerical integrations.

Excitation control of a power-generating system based on fuzzy logic and neural networks

This paper presents a practical design of an intelligent controller (using fuzzy logic and neural network concepts) for the excitation control of an isolated power-generating system. The controller is suitable for realtime operation, with the aim of improving the dynamic characteristics of the generating unit by acting properly on the exciter input. At first, digital simulations of high- and reduced-order models of the above system are performed using conventional control techniques on five system cases which are based on previous work. Then, a fuzzy logic proportional-plus-derivative controller (FPDC) is designed and the dynamic performances of the mentioned five system cases and the FPDCs are presented by comparison. Finally, an enhanced controller is designed which involves a pre-trained neural network as a model dynamics capture. The computer simulation results obtained clearly demonstrate that the performance of the developed fuzzy logic and neural network controller offers better damping effects on the generator oscillations over a wider range of operating conditions, than the associated ones of the conventional excitation controller designs with output feedback applied to high- and reduced-order system models.

Component Mode Synthesis Using Nonlinear Normal Modes

This paper describes a methodology for developing reduced-order dynamic models of structural systems that are composed of an assembly of nonlinear component structures. The approach is a nonlinear extension of the fixed-interface component mode synthesis (CMS) technique developed for linear structures by Hurty and modified by Craig and Bampton. Specifically, the case of nonlinear substructures is handled by using fixed-interface nonlinear normal modes (NNMs). These normal modes are constructed for the various substructures using an invariant manifold approach, and are then coupled through the traditional linear constraint modes (i.e., the static deformation shapes produced by unit interface displacements). A class of systems is used to demonstrate the concept and show the effectiveness of the proposed procedure. Simulation results show that the reduced-order model (ROM) obtained from the proposed procedure outperforms the ROM obtained from the classical fixed-interface linear CMS approach as applied to a nonlinear structure. The proposed method is readily applicable to large-scale nonlinear structural systems that are based on finite-element models.

Aeroelastic System Development Using Proper Orthogonal Decomposition and Volterra Theory

This research combines Volterra theory and proper orthogonal decomposition (POD) into a hybrid methodology for reduced-order modeling of aeroelastic systems. The outcome of the method is a set of linear ordinary difierential equations (ODEs) describing the modal amplitudes associated with both the structural modes and the POD basis functions for the ∞uid. For this research, the structural modes are sine waves of varying frequency, and the Volterra-POD approach is applied to the ∞uid dynamics equations. The structural modes are treated as forcing terms which are impulsed as part of the ∞uid model realization. Using this approach, structural and ∞uid operators are coupled into a single aeroelastic operator. This coupling converts a free boundary ∞uid problem into an initial value problem, while preserving the parameter (or parameters) of interest for sensitivity analysis. The approach is applied to an elastic panel in supersonic cross ∞ow. The hybrid Volterra-POD approach provides a low-order ∞uid model in state-space form. The linear ∞uid model is tightly coupled with a nonlinear panel model using an implicit integration scheme. The resulting aeroelastic model provides correct limit-cycle oscillation prediction over a wide range of panel dynamic pressure values. Time integration of the reduced-order aeroelastic model is four orders of magnitude faster than the high-order solution procedure developed for this research using traditional ∞uid and structural solvers.

The Matrix Pencil for Power System Modal Extraction

This work introduces the matrix pencil modal extraction method through the use of several illustrative examples. This method is used to estimate the eigenvalues of reduced-order models of large nonlinear systems based on their dynamic responses.

Characterisation of rotating machinery dynamics within limited frequency intervals using modal analysis

The possibility of characterising the dynamics of rotating machinery structures using modal analysis is explored. The paper is based on a recently-developed method by which the left-eigenvectors can be computed from their right counterparts and their corresponding eigenvalues. This method is aimed at obtaining the complete set of modal parameters of a machine without the need to measure a complete row plus a complete column of the frequency response function (FRF) matrix, as is customary for non-self-adjoint systems. The applicability of the method to the derivation of reduced-order modal models of continuous systems from measured FRFs is discussed. It is shown that the new method is applicable if the response of the continuous system at hand is only affected significantly by a limited number of modes within the frequency interval of interest. It is also shown that a simplification of the measurement scheme made available by this new method depends on the selection of degrees of freedom used to study the structure. A numerical example is given to illustrate this point.

Suboptimal model reduction for singular systems

The model reduction problem was investigated for singular systems in this paper. To solve the problem, the Silverman–Ho algorithm was given, and based on this algorithm, an optimal model reduction method for the fast subsystem was presented to obtain reduced-order stable models for singular systems. Sequentially, the suboptimal model reduction algorithm was obtained for singular systems. The advantage of the presented algorithm is that the impulsive nature of singular system is preserved in the reduced-order models. Then some necessary and sufficient conditions for the existence of a stable reduced-order system were given and these conditions can be verified numerically. Finally, illustrative examples were given to show the effectiveness of the proposed approach.