Model Reduction Methodology Research Articles

MODEL REDUCTION OF NONLINEAR SYSTEMS: A GREY-BOX MODELING APPROACH1

We present a novel model reduction methodology for the approximation of large-scale nonlinear systems. The methodology stems from the need to find computationally efficient substitute models for nonlinear systems. The nonlinear system is viewed as a grey-box model with a mechanistic (first-principle) component and an empirical (black-box) component identified for the computationally intensive parts of the nonlinear system. The mechanistic part is approximated using proper orthogonal decompositions whereas the empirical part is identified as polynomial functions by parameter estimation using the reduced order mechanistic part.

Reduced computations for nematic-liquid crystals: A timestepper approach for systems with continuous symmetries

In this work we address the development of a timestepper based computational approach to analyse the dynamics of the Smoluchowski equation directly, bypassing the necessity of finding closures to construct low-order macroscopic models, while at the same time factoring out in a dynamic way the underlying continuous symmetry. This can be thought as a two-tier model reduction methodology for systems with continuous symmetries: going from models with a large numbers of degrees of freedom to more compact models, in terms of a smaller number of dependent variables, while also taking care of the underlying invariance.

Modelling and reduction techniques for studies of integrated hybrid vehicle systems

Models of integrated vehicle systems are essential for designing hybrid vehicles by means of simulation-based optimization. Given the complexity of hybrid vehicle systems, designing is a time consuming process that requires the evaluation of a large number of different design configurations. Modelling and simulation can significantly reduce the design time through efficient design evaluations and reduced number of prototypes built. This work presents the development and reduction of an integrated hybrid vehicle model composed of an engine, drivetrain, hydraulics and vehicle dynamics subsystems. For model development the bond graph formulation is used because it facilitates the integration of component/subsystem models in different energy domains, supports hierarchical modelling and allows straightforward manipulation of the model. The model is configured for a medium size military truck, and implemented in the 20SIM modelling and simulation environment. After developing the model, an energy-based model reduction methodology is applied in order to generate a reduced vehicle model that provides more design insight. The generated reduced system model for the hybrid truck (compared to the full model) produces almost identical predictions, has almost half the size and calculates the system response 2.5 times faster. This computationally efficient reduced model can be used for vehicle design studies to further reduce the development time.

Generating proper dynamic models for truck mobility and handling

In previous work, a proper was defined as the model of minimal complexity, with physically meaningful parameters, which accurately predicts dynamic system outputs. Proper models can be generated by an energy-based model reduction methodology that removes unnecessary complexity from models (linear or nonlinear) without altering the physical interpretation of the remaining parameters and variables. Energy-based model reduction allows design in a search space of reduced dimension, and in general improves computational efficiency. The current work demonstrates the effectiveness and utility of an energy-based model reduction algorithm for vehicle system modelling applications. Model reduction is performed for two different vehicle modelling applications. The first case study focuses on the vehicle dynamics model of a military heavy-duty tractor semi-trailer. The full model is developed using a classical multibody system approach and accurate reduced models are then generated for a lane change manoeuvre. The second case study develops, validates, and reduces an integrated vehicle system model of a single-unit medium-size commercial truck composed of engine, drivetrain, and vehicle dynamic subsystems. The systematically reduced model accurately predicts vehicle forward speed/acceleration and engine behaviour during full-throttle acceleration and braking with a twofold increase in computation efficiency. The reduced models generated by the energy-based methodology retain predictive quality, are useful for studying trade-offs involved in redesigning components and control strategies for improved vehicle performance, and are less computationally intensive.

Frequency-Domain-Based, Control-Relevant Model Reduction for Nonlinear Plants

Chemical plants exhibit nonlinear dynamics that need to be captured accurately for model-based control. This paper focuses on a frequency-domain-based, control-relevant model reduction strategy for nonlinear process plants. The generalized frequency response function proposed for the nonlinear plants has been used to analyze the characteristics of nonlinear structures in the frequency domain, from a control-relevant viewpoint. Control-relevant model reduction is sought by minimizing the mismatch between a higher order, nonlinear proxy plant and candidate reduced order nonlinear model structures. The minimization is proposed to be done by prefiltering of identification data to shape the mismatch in a control-relevant manner. For these prefilters as well, candidate linear and nonlinear structures are evaluated and an algorithm to design the prefilters to reflect the servo and regulatory performance specifications is proposed. The model reduction methodology is used along with a nonlinear internal model control (NIMC) algorithm and is demonstrated on a simple simulation as well as on a representative nonlinear chemical reactor studied in the literature.

Nonlinear dynamics and control of process systems with recycle

Abstract Process systems with material and energy recycle are well-known to exhibit complex dynamics and to present significant control challenges, due to the feedback interactions induced by the recycle streams. In this paper, we address the dynamic analysis and control of such process systems. Initially, we establish, through an asymptotic analysis, that (i) small recycle flowrates induce a weak coupling among individual processes, whereas (ii) large recycle flowrates induce a time scale separation, with the dynamics of individual processes evolving in a fast time scale with weak interactions, and the dynamics of the overall system evolving in a slow time scale where these interactions become significant; these slow dynamics is usually nonlinear and of low order. Motivated by this, we present (i) a model reduction methodology for deriving nonlinear low-order models of the slow dynamics induced by large recycle streams, and (ii) a controller design framework consisting of properly coordinated controllers in the fast and the slow time scales. The theoretical results are illustrated in a reaction-separation network with a large recycle compared to the throughput.

Nonlinear Dynamics and Control of Process Systems with Recycle

Process systems with material and energy recycle are well-known to exhibit complex dynamics and to present significant control challenges, due to the feedback interactions induced by the recycle streams. In this paper, we address the dynamic analysis and control of such process systems. Initially, we establish, through an asymptotic analysis, that i) small recycle flowrates induce a weak coupling among individual processes, whereas ii) large recycle flowrates induce a time scale separation, with the dynamics of individual processes evolving in a fast time scale with weak interactions, and the dynamics of the overall system evolving in a slow time scale where these interactions become significant; this slow dynamics is usually nonlinear and of low order. Motivated by this, we present i) a model reduction methodology for deriving nonlinear low-order models of the slow dynamics induced by large recycle streams, and ii) a controller design framework comprising of properly coordinated controllers in the fast and the slow time scales. The theoretical results are illustrated in a reaction-separation network with a large recycle compared to the throughput.

Time Domain Finite Element Analysis of Dynamic Systems

In this study, dynamic analyses of systems are conducted using the time domain finite element analysis when initial and final conditions are specified. Dynamics of systems can be analyzed using the time-marching procedure based on Hamilton's weak principle. A new method is developed to describe the motion of a system and is extended to accommodate the initial displacement as a state variable. System matrices are constructed to evaluate time histories of the system from initial time to a specified time without the information on the velocity and momentum at each time step. Galerkin's weak principle is used to construct element matrices in the time domain and assemble the whole matrices. The Neumann boundary condition is modified to reflect the effect of initial displacement and velocity that are inherent to dynamic problems through the momentum conservation of the system. The matrices obtained thereafter conserve the symmetricity and enable us to reduce the model order by the systematic approach. Spatial propagation equation is built up, and two-point boundary conditions are used to estimate the unknown initial conditions at one end of the beam. Modal domain analysis is introduced to reduce the size of matrices. It can be seen that the matrices constructed using the time domain finite element analysis are applicable to the model reduction that is analogous to the space-domain model reduction methodology. Several numerical examples show that a consequence of the suggested method can be applied to describe the motion of various dynamic systems successfully and to validate the effect of the model reduction in the time domain.

Model reduction for optimization of rapid thermal chemical vapor deposition systems

A model of a three-zone rapid thermal chemical vapor deposition (RTCVD) system is developed to study the effects of spatial wafer temperature patterns on polysilicon deposition uniformity. A sequence of simulated runs is performed, varying the lamp power profiles so that different wafer temperature modes are excited. The dominant spatial wafer thermal modes are extracted via proper orthogonal decomposition and subsequently used as a set of trial functions to represent both the wafer temperature and deposition thickness. A collocation formulation of Galerkin's method is used to discretize the original modeling equations, giving a low-order model which loses little of the original, high order model's fidelity. We make use of the excellent predictive capabilities of the reduced model to optimize power inputs to the lamp banks to achieve a desired polysilicon deposition thickness at the end of a run with minimal deposition spatial nonuniformity. Since the results illustrate that the optimization procedure benefits from the use of the reduced-order model, our future goal is to integrate the model reduction methodology into real-time and run-to-run control algorithms. While developed in the context of optimizing a specific RTP process, the model reduction techniques presented in this paper are applicable to other materials processing systems.

Component model reduction methodology for articulate multiflexible body structures

Component model reduction methodology for articulate multiflexible body structures

Model reduction methodology for articulated, multiflexible body structures

Multibody dynamics simulation packages such as DISCOS are commonly used to simulate and analyze systems of interconnecte d rigid or flexible bodies. In many applications using these codes, the flexible components must be described in a way that when they are assembled, important system-level modes are accurately captured. The projection and assembly model reduction method, developed at the Jet Propulsion Laboratory, serves this purpose very well. However, early application of the method on the Galileo spacecraft model indicated mismatches in transfer function zeros between the full-order model and a reduced-order model constructed using the projection and assembly method. In this paper, an enhanced projection and assembly method, employing static correction modes to augment either the selected mode set of the system or the projected mode sets of the components is presented. The effectiveness of the proposed methodology in generating reduced-order system models, accurate at multiple system configurations and over a frequency range of interest, has been successfully demonstrated on a high-order finite element model of the Galileo spacecraft.

Component modes damping assignment methodology for articulated, multiflexible body structures

To simulate the dynamical motion of articulated, multiflexible body structures, one can use multibody simulation packages such as DISCOS. To this end, one must supply appropriate reduced-order models for all of the flexible components involved. The component modes projection and assembly model reduction (COMPARE) methodology is one way to construct these reduced-order component models, which when reassembled capture important system input-to-output mapping of the full-order model at multiple system configurations of interest. In conjunction, we must also supply component damping matrices which when reassembled generate a system damping matrix that has certain desirable properties. The problem of determining the damping factors of components' modes to achieve a given system damping matrix is addressed here. To this end, we must establish from first principles a matrix-algebraic relation between the system's modal damping matrix and the components' modal damping matrices. An unconstrained/constrained optimization problem can then be formulated to determine the component modes' damping factors that best satisfy that matrix-algebraic relation. The effectiveness of the developed methodology, called ModeDamp, has been successfully demonstrated on a high-order, finite element model of the Galileo spacecraft.

Model reductions using a projection formulation

A new methodology for model reduction of multi-input multi-output systems exploits the notion of an oblique projection. A reduced model is uniquely defined by a projector whose range space and orthogonal to the null space are chosen among the ranges of generalized controllability and observability matrices. The reduced order models match various combinations (chosen by the designer) of four types of parameters of the full order system associated with (i) low frequency response, (ii) high frequency response, (iii) low frequency power spectral density and (iv) high frequency power spectral density. Thus, the proposed method is a computationally simple substitute for many existing methods, has an extreme flexibility to embrace combinations of existing methods and offers some new features.