MIMO Model Order Reduction by Padé–Markov Least Squares and Frequency Data Fitting

We discuss methods for model order reduction (MOR) of linear systems with many input and output variables, arising in the modeling of linear (sub) circuits with a huge number of nodes and a large number of terminals, like power grids. Our work is based on the approaches SVDMOR and ESVDMOR proposed in recent publications (1; 2; 3; 4; 5). In particular, we discuss efficient numerical algorithms for their implementation. Only by using efficient tools from numerical linear algebra, these methods become applicable for truly large-scale problems.

The state-space modeling and related order reduction problems of switching DC-DC converters operating in discontinuous (D) mode are treated in this paper. A unified discrete-time full-order model is proposed assuming the discontinuous current and voltage mode (DCVM) as a general operation mode from which the discontinuous current mode (DCM), the discontinuous voltage mode (DVM), and continuous made (CM) derive as special cases. Such a model permits a straightforward study of the control-to-output response, in each operation mode. The order reduction of the transfer function is carried out by means of the principal component analysis considering the joint controllability and observability properties of the circuit. It is shown that a proper order reduction can be gained not only in DC mode, as expected, but also in DV and DCV modes. A substantial difference between DC and DV modes is also evidenced in terms of controllability. The frequency analysis and state-space model order reduction for a Cuk converter are proposed to show that the method is free of the limitations affecting other methods of analysis and that its predictions fit experimental measurements.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow

In this paper a new model order reduction of linear continuous time system is proposed. An improved technique of Intelligent Search Evolution algorithm (ISEA) is used for generating the reduced order model. The proposed order reduction technique have the advantages of obtaining optimal model order reduction with less no of iterations and faster convergence with less computational time. The feasibility of proposedmethodis explained with the help of Himmelblau function and model order reduction. The efficacy of the proposed order reduction technique is illustrated through a numerical example and results are compared with other existing model order reduction technique. The reduced order model generated by using ISEA will maintain the characteristics of original system very closely.

Previous work by the authors developed algorithms for simplifying the structure of a lumped dynamic system model and reducing its order. This paper extends this previous work to enable simultaneous model structure and order reduction. Specifically, it introduces a new energy-based metric to evaluate the relative importance of energetic connections in a model. This metric (1) accounts for correlations between energy flow patterns in a model using the Karhunen-Loe`ve expansion; (2) examines all energetic connections in a model, thereby assessing the relative importance of both energetic components and their interactions, and enabling both order and structural reduction; and (3) is realization-preserving, in the sense of not requiring a state transformation. A reduction scheme based on this metric is presented and illustrated using a simple example.

A computer code capable of reliably predicting an optimal control strategy for a complete pressurized water reactor nuclear power plant is developed. The state, control, disturbance, and output vectors for the reactor, steam generator, pressurizer, reactor coolant pump, and turbogenerator unit are defined. The overall system is found to be of order 38, and calculational management requires an order reduction. This is accomplished using the Litz method of dominance measures. Order reduction based solely on the location of the eigenvalues in the s plane is also investigated. For the same reduced-order model, the Litz method dominance hierarchy of the plant states seems to make better sense, and unlike the conventional eigenvalue approach it gives accurate results when compared to original system responses. Finally, an overall plant observer is designed based on a seventeenth reduced-order system model. It is based on greatly reduced calculations, thus requiring less hardware, and is a more reliable predictor.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Black-box system identification for reduced order model construction

With the increasing need for flexible operation (shorter startup time and higher load change rate etc.), clearance monitoring between the rotor and the stationary components in steam turbines is becoming more important. This is because as load change rate increases, minimum radial and axial clearances during operation tend to be smaller due to thermal deformation of steam turbines, and the risk of contact between the rotor and the stationary components becomes higher. This situation has accelerated development of clearance sensors. However, it is still difficult to monitor all possible points of contact only with physical sensors due to limited installation location and short lifetime in high temperature environment. From the above background, we have been developing a virtual clearance monitoring (VCM) technique based on a novel data fusion approach that utilizes both physical and data-driven models. Specifically, a reduced order model (ROM) is used as physical model in order to enable real-time prediction with an accuracy similar to that of finite element analysis (FEA). Then, the prediction error of the physical model is corrected by using a residual model built by machine learning from the past clearance sensor values and the corresponding physical model-based prediction results. As will be explained in this report, this technique has an advantage that the clearances can be predicted in real-time based only on operating data such as steam conditions at inlet and outlet, and some temperatures in the parts not modeled in the ROM. Therefore, the virtual sensor based on this technique can be used as a replacement for the physical sensor after it has failed. Furthermore, this technique can also be used to preliminarily study unsteady clearance behavior for inexperienced operating conditions. This paper describes how to build the ROM from a finite element model for thermal-structural analysis of an entire steam turbine by model order reduction (MOR), and the detail of the VCM technique, and a VCM system installed in a measurement room of a state-of-the-art GTCC power plant manufactured by Mitsubishi Power. In addition, the verification results of the VCM system are presented. In this research, the ROM and the residual model were built using the data obtained from four operations with different start-up modes each other. Then, VCM was performed for 12 operating cases. As a result, this survey revealed the followings: (1) This system is capable of real-time prediction with output intervals of roughly 2 seconds. (2) As for radial clearance prediction error during rotor rotating, the RMSEs and the absolute values of minimum value errors are less than or equal to 7.2 % and 7.0 % respectively relative to an initial radial clearance value during the steam turbine stopping. From the above results, we conclude that this VCM technique based on data fusion approach is effective in terms of computational speed and prediction accuracy. This means that if a physical clearance sensor fails, the radial clearance can be continuously monitored by a virtual clearance sensor with a residual model built using the data when the sensor was working normally. In the future, we plan to further improve the accuracy of this technique through improvement in physical modeling.

&lt;p&gt;Order reduction of linear-time invariant systems employing two methods; one using the advantages of Routh approximation and other by an evolutionary technique is presented in this paper. In Routh approximation method the denominator of the reduced order model is obtained using Routh approximation while the numerator of the reduced order model is determined using the indirect approach of retaining the time moments and/or Markov parameters of original system. By this method the reduced order model guarantees stability if the original high order model is stable. In the second method Particle Swarm Optimization (PSO) is employed to reduce the higher order model. PSO method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. Both the methods are illustrated through numerical examples.&lt;/p&gt;

A model order reduction method is proposed for models of aeroservoelastic vehicles in the linear parameter-varying (LPV) systems framework, based on state space interpolation of modal forms. The dynamic order of such models is usually too large for control synthesis and implementation since they combine rigid body dynamics, structural dynamics and unsteady aerodynamics. Thus, model order reduction is necessary. For linear timeinvariant (LTI) models, order reduction is often based on balanced realizations. For LPV models, this requires the solution of a large set of linear matrix inequalities (LMIs), leading to numerical issues and high computational cost. The proposed approach is to use well developed and numerically stable LTI techniques for reducing the LPV model locally and then to transform the resulting collection of systems into a consistent modal representation suitable for interpolation. The method is demonstrated on an LPV model of the body freedom flutter vehicle, reducing the number of states from 148 to 15. The accuracy of the reduced order model (ROM) is confirmed by evaluating the ν-gap metric with respect to the full order model and by comparison to another ROM obtained by state-of-the-art LPV balanced truncation techniques.

Efficient full-chip thermal simulation is among the most challenging problems facing the EDA industry today, due to the need for solution of very large systems of equations that require unreasonably long computational times. However, in most cases, temperature is not required to be computed at every point of the IC but only at certain hotspots, in order to assess the circuits compliance with thermal specifications. This makes the thermal analysis problem amenable to Model Order Reduction (MOR) techniques. System-theoretic techniques like Balanced Truncation (BT) offer very reliable bounds for the approximation error, which can be used to control the order and accuracy of the reduced models during creation, at the expense of greater computational complexity to create them. In this paper, we propose a computationally efficient low-rank BT algorithm that retains all the system-theoretic advantages in the reduction of model order for fast hotspot thermal simulation. Experimental results demonstrate a 39X order reduction and very tight accuracy bounds.

On proper orthogonal decomposition (POD) based reduced-order modeling of groundwater flow through heterogeneous porous media with point source singularity

The reduction of the large order models of complex systems into a sufficiently low dimensional order in such a way that important dynamic characteristics of the real system are preserved, is a nontrivial task. The motivations for such a model order reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. In the present article, major techniques in model order reduction are briefly discussed. The characteristic similarities and differences are highlighted, their advantages and disadvantages are underlined, and further research requirements and trends are commented upon. Introduction and Perspective One of the central issues in the active control of complex systems such as large flexible space structures (LFSS) is the derivation of a correct mathematical model for both the controlled and the uncontrolled dynamical systems. Theoretically, there are infinitely many elastic modes or degrees of freedom (DOF) in the distributed parameter (DP) models of LFSS, usually with very low natural damping. Moreover, the flexible modes contribute to the actual deformation of the structure.' The truly infinite dimensional character of LFSS models has to be approximated by some fidelity finite (but usually very large) dimensional model. The normal approach taken by engineers to achieve this end is via modal models with a large number of modes that provide a reasonable representation of the spacecraft dynamic characteristics. However, a difficult problem still remains to be the development of a model of lowenough dimensional order that it can be utilized by the onboard controller, yet high enough dimensional order that it preserves the dynamic characteristics of the real system represented and controlled. The motivations for such a reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. Discretization procedures will not be discussed herein and it will be assumed that a large finite dimensional model has been generated somehow and for implementation and other practical considerations, the model needs to be further reduced. *This work was-supported, in part, by the Air Force Wright Aeronautical, Air Force Systems Command, Flight Dynamics Lab, Structures and Dynamics Division, while the author was employed with HR-Textron, Valencia, California. There are various approaches to model order reduction; some are or pseudo-optimal and others are ad-hoc methods based on practical considerations and engineering experience and judgment. These techniques are known by different names such as reduction, condensation, economization, aggregation, optimal projection, and other combinations thereof. It has often been pointed out that such techniques generally constitute application of Rayleigh-Ritz/Galerkin optimization, and matrix transformation methods to the eigenvalue/eigenvector formulation for structural dynamic problems. The reduction or the condensation methods are based on transformations of the coordinates in the equations of motion that essentially maintain the invariance of the quadratic forms of the potential and kinetic energies. An important feature of these methods is that the reduced order model often loses the basic characteristics of the original system. Considerable progress was made by Likins, Ohkami, and wong5 in this respect. However, they failed to develop a general enough criterion that could reduce the system model in an sense without significantly affecting the eigenvalues of the original model. The optimization or the mathematical reduction procedure, on the other hand, is based on the reduction of the eigenvalue problem to a smaller size based on some optimality criterion (usually quadratic). In these techniques, the reductions are carried out without actually resorting to approximations, i .e. , without truncatiag any coordinates or states of the original system. Balanced model reduction of linear timeinvariant dynamical systems is essentially based on the controllability and observability relations of the states of the system. Subsystem models are obtained by deleting those states that contribute the least to the controllability and observability (or the impulse response) of the original system7 and thus are only in this sense. A more recent approach to model reduction is proposed in Skelton's work,* where each state of the system model is assigned a relative to a given basis, via a quadratic criterion, and the states with the least cost are deleted in a systematic manner. The resulting reduced model is a function of the state-space basis, and thus there is no guarantee for optimality for all choices. The latest development in model order reduction techniques is the work by Hyland and erns stein.^ Herein first order necessary conditions for reduced order modeling of linear time-invariant systems are derived via a pair of modified Lyapunov equations coupled by a nonorthogonal projection. This approach reveals the possibility of multiple extrema forsome of the abovementioned methods. Copyright @ 1988 by the American Institute of Aeronautics and Astronautics. Inc. All Rights Reserved.

In recent years, autonomously guided parachute systems have been developed with the goal of increasing the landing accuracy of airdrop operations. These precision airdrop systems utilize an onboard global positioning system (GPS) coupled with a guidance, navigation and control (GNC) system to continuously monitor and correct the flight path of the parachute system during descent. A critical component of the GNC system is an algorithm that predicts the response of the system to various aerodynamic forces in real-time that are typically based on a low-order multi-degree of freedom (DOF) dynamics model. Reduced order modeling is the process of representing a complex dynamic system with many DOF with a model having relatively few DOF. The goal of this study is to examine the use of large scale finite element models to construct and calibrate Reduced Order Models (ROM) for parachute systems. The resulting ROM would provide parachute designers with a simple tool to examine parachute behavior under various operating conditions. The use of numerical simulations to calibrate the ROM would help to eliminate the number of physical tests needed to characterize a parachute system’s properties. In this study, a 6 degree of freedom ROM is adopted. The system dynamic properties for the ROM and the resultant forces and moments acting on the ROM are obtained from finite element simulations. The 6 DOF model originally proposed by Cockrell and Doherr and later used by Dobrokhodov is adopted for the ROM. The finite element (FE) code, TENSION, is used to perform the large scale simulations and extract the ROM dynamic properties and resultant forces. Several examples are performed that verify the ROM procedure by comparing ROM predictions to FE simulations for various parachute system trajectories. The performance of the ROM is shown to be excellent for in-plane motion under an oscillating wind field. It is shown, however, that the ROM procedure deteriorates as the FE model is refined by adding more DOF. It is demonstrated that, for highly deformable FE models with many DOF, more sophisticated ROM procedures are needed.

Non-linear analysis is of increasing importance in wind energy engineering as a result of their exposure in extreme conditions and the ever-increasing size and slenderness of wind turbines. Whilst modern computing capabilities facilitate execution of complex analyses, certain applications which require multiple or real-time analyses remain a challenge, motivating adoption of accelerated computing schemes, such as reduced order modelling (ROM) methods. Soil structure interaction (SSI) simulations fall in this class of problems, with the non-linear restoring force significantly affecting the dynamic behaviour of the turbine. In this work, we propose a ROM approach to the SSI problem using a recently developed ROM methodology. We exploit a data-driven non-linear ROM methodology coupling an autoencoder with long short-term memory (LSTM) neural networks. The ROM is trained to emulate a steel monopile foundation constrained by non-linear soil and subject to forces and moments at the top of the foundation, which represent the equivalent loading of an operating turbine under wind and wave forcing. The ROM well approximates the time domain and frequency domain response of the Full Order Model (FOM) over a range of different wind and wave loading regimes, whilst reducing the computational toll by a factor of 300. We further propose an error metric for capturing isolated failure instances of the ROM.