Model Approximation Problem Research Articles

Blackbox optimization for approximating high-fidelity heat transfer calculations in metal additive manufacturing

Heat transfer simulations play an important role in predicting solidification conditions that determine microstructure properties of additive manufactured products. High-fidelity heat transfer models, e.g., Truchas [1], provide a numerical framework capable of resolving many desired physics (e.g., thermal radiation, elemental vaporization, gas-solid-liquid interactions, etc.), but they are often too computationally expensive to simulate the solidification conditions at the AM process-scale. Recently, Stump et al. [2] proposed a low-fidelity analytical heat conduction model capable of simulating solidification conditions on the process-scale. This model makes the problem analytically tractable by neglecting many important physics that may have an impact on the accuracy of the predicted solidification conditions. An effective way of improving the accuracy of the analytical with little computational effort is to determine an optimal set of parameters that minimize the discrepancy between the low-fidelity analytical model and a high-fidelity numerical model. In this work, a model approximation problem is formulated which can be solved by leveraging blackbox optimization methods. Multiple heat sources are used in the analytical model to improve the calibration performance in a flexible framework. Two blackbox optimization methods, i.e., Bayesian optimization and directional Gaussian smoothing methods, are employed to address the challenge that the gradient of the loss function is inaccessible during optimization. The results show that the single-beam analytical model has limitations in fitting the high-fidelity model, but the proposed multi-beam analytical model provides satisfactory approximations to the high-fidelity temperature and melt pool fields.

Optimal Hankel norm model reduction for discrete-time descriptor systems

Optimal Hankel norm model reduction for dynamical systems is of great significance in model-based simulation and design. For the class of linear time-invariant systems, it is among the few optimal reduction methods for which a prior error bound between the original system and its approximation is known. However, for descriptor systems, this optimal approximation technique no longer applies. In this paper, we propose several definitions of the Hankel operator for dynamical discrete-time descriptor systems. We investigate the implications of these definitions for the problem of optimal model approximation of descriptor systems in the sense of the Hankel norm. Novel reduction algorithms are derived for this class of systems with and without preservation of the DAE-index. The performance of the proposed methods is illustrated by numerical examples.

Model Approximation for Switched Genetic Regulatory Networks.

The model approximation problem is studied in this paper for switched genetic regulatory networks (GRNs) with time-varying delays. We focus on constructing a reduced-order model to approximate the high-order GRNs considered under the switching signal subject to certain constraints, such that the approximation error system between the original and reduced-order systems is exponentially stable with a disturbance attenuation performance. The stability conditions and the disturbance attenuation performance are established by utilizing two integral inequality bounding techniques and the average dwell-time method for the approximation error system. Then, the solvability conditions for the reduced-order models for the GRNs are also established using the projection method. Furthermore, the model approximation problem can be transferred into a sequential minimization problem that is subject to linear matrix inequality constraints by using the cone complementarity algorithm. Finally, several examples are provided to illustrate the effectiveness and the advantages of the proposed methods.

Reduced-order model approximation of fuzzy switched systems with pre-specified performance

The reduced-order model approximation problem for discrete-time hybrid switched nonlinear systems is addressed via Takagi-Sugeno (T-S) fuzzy modelling in this paper. For a high-dimension hybrid switched nonlinear system, our aim is on how to construct a reduced-dimension hybrid switched model, approximating its original high-order model well with a pre-specified system performance level. Firstly, the mean-square exponential stability analysis is provided, in which it guarantees the given weighted H∞ system performance level for the augmented error dynamic system by the average dwell time analysis approach and the hybrid switched Lyapunov stability theory. The solution of corresponding model reduction problem with pre-specified performance is given by using the projection Lemma. The algorithm of the reduced-order hybrid switched model parameters is designed by the cone complementary linearization technique. Finally, the advantage and effectiveness of the proposed reduced-order model approximation approach are shown by the simulation result.

Resilient model approximation for Markov jump time-delay systems via reduced model with hierarchical Markov chains

ABSTRACTIn this paper, the resilient model approximation problem for a class of discrete-time Markov jump time-delay systems with input sector-bounded nonlinearities is investigated. A linearised reduced-order model is determined with mode changes subject to domination by a hierarchical Markov chain containing two different nonhomogeneous Markov chains. Hence, the reduced-order model obtained not only reflects the dependence of the original systems but also model external influence that is related to the mode changes of the original system. Sufficient conditions formulated in terms of bilinear matrix inequalities for the existence of such models are established, such that the resulting error system is stochastically stable and has a guaranteed l2-l∞ error performance. A linear matrix inequalities optimisation coupled with line search is exploited to solve for the corresponding reduced-order systems. The potential and effectiveness of the developed theoretical results are demonstrated via a numerical example.

Model Approximation for Fuzzy Switched Systems With Stochastic Perturbation

In this paper, the model approximation problem is investigated for a Takagi–Sugeno fuzzy switched system with stochastic disturbance. For a high-order considered system, our attention is focused on the construction of a reduced-order model, which not only approximates the original system well with a Hankel-norm performance but translates it into a lower dimensional fuzzy switched system as well. By using the average dwell time approach and the piecewise Lyapunov function technique, a sufficient condition is first proposed to guarantee the mean-square exponential stability with a Hankel-norm error performance for the error system. The model approximation is then converted into a convex optimization problem by using a linearization procedure. Finally, simulations are provided to illustrate the effectiveness of the proposed theory.

Pre-specified performance based model reduction for time-varying delay systems in fuzzy framework

This paper attempts to provide a new solution to the model approximation problem for dynamic systems with time-varying delays under the fuzzy framework. For a given high-order system, our focus is on the construction of a reduced-order model, which approximates the original one in a prescribed error performance level and guarantees the asymptotic stability of the corresponding error system. Based on the reciprocally convex technique, a less conservative stability condition is established for the dynamic error system with a given error performance index. Furthermore, the reduced-order model is eventually obtained by applying the projection approach, which converts the model approximation into a sequential minimization problem subject to linear matrix inequality constraints by employing the cone complementary linearization algorithm. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

Formula omitted] model approximation for discrete-time Takagi–Sugeno fuzzy systems with Markovian jumping parameters

This paper is concerned with the H∞ model approximation problem for a class of discrete-time Takagi–Sugeno (T–S) fuzzy Markov jump systems. The systems involve stochastic disturbances and nonlinearities that can be described by T–S fuzzy models. The problem to be solved in the paper is to find a reduced-order model, which is able to approximate the original T–S fuzzy Markov jump system with comparatively small and acceptable errors. Specifically, the corresponding error system is guaranteed to be asymptotically stable in the mean square with a prescribed H∞ performance index. By using convex optimization approach and projection approach, respectively, sufficient conditions on the existence for such model with reduced-order are obtained and presented in the form of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of the obtained results.

Time-Delay System Identification Using Genetic Algorithm – Part Two: FOPDT/SOPDT Model Approximation

Abstract The First-Order-Plus-Dead-Time (FOPDT) or Second-Order-Plus-Dead-Time (SOPDT) model approximation to a complicated process system can be carried out through either a kind of model reduction approach or a kind of system identification approach. This paper investigates this model approximation problem through an identification approach using the real coded Genetic Algorithm (GA). The desired FOPDT/SOPDT model is directly identified based on the measured system's input and output data. In order to evaluate the quality and performance of this GA-based approach, the proposed method is compared with two typical model reduction methods, namely Skogestad's rules and Sung et al method. The obtained results exhibit a very promising capability of GA in handling the data-driven time-delay system approximation.

Model Approximation for Discrete-Time State-Delay Systems in the T–S Fuzzy Framework

This paper is concerned with the problem of H∞ model approximation for discrete-time Takagi-Sugeno (T-S) fuzzy time-delay systems. For a given stable T- S fuzzy system, our attention is focused on the construction of a reduced-order model, which not only approximates the original system well in an H∞ performance but is also translated into a linear lower dimensional system. By applying the delay partitioning approach, a delay-dependent sufficient condition is proposed for the asymptotic stability with an H∞ error performance for the error system. Then, the H∞ model approximation problem is solved by using the projection approach, which casts the model approximation into a sequential minimization problem subject to linear matrix inequality (LMI) constraints by employing the cone complementary linearization algorithm. Moreover, by further extending the results, H∞ model approximation with special structures is obtained, i.e., delay-free model and zero-order model. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods.

Generalized ℋ︁2 model approximation for differential linear repetitive processes

This paper investigates the generalized ℋ︁2 model approximation for differential linear repetitive processes (LRPs). For a given LRP, which is assumed to be stable along the pass, we are aimed at constructing a reduced-order model of the LRP such that the generalized ℋ︁2 gain of the approximation error LRP between the original LRP and the reduced-order one is less than a prescribed scalar. A sufficient condition to characterize the bound of the generalized ℋ︁2 gain of the approximation error LRP is presented in terms of linear matrix inequalities (LMIs). Two different approaches are proposed to solve the considered generalized ℋ︁2 model approximation problem. One is the convex linearization approach, which casts the model approximation into a convex optimization problem, while the other is the projection approach, which casts the model approximation into a sequential minimization problem subject to LMI constraints by employing the cone complementary linearization algorithm. A numerical example is provided to demonstrate the proposed theories. Copyright © 2010 John Wiley & Sons, Ltd.

Delay-dependent and delay-independent energy-to-peak model approximation for systems with time-varying delay

This paper deals with the problem of computing an approximation system for a given system with time-varying delay such that the energy-to-peak gain of the error system is less than a prescribed scalar. First, a delay-dependent boundedness condition of energy-to-peak gain is given in terms of linear matrix inequalities (LMIs), which recovers the delay-independent case. Then, based on the established delay-dependent boundedness condition of energy-to-peak gain, a sufficient condition to characterize the approximation system is obtained to solve the energy-to-peak model approximation problem in the form of LMIs with inverse constraints. A number of delay-independent energy-to-peak model approximation cases are special cases of a delay-dependent approximation. An efficient algorithm is derived to obtain the approximation models. Finally, examples are employed to demonstrate the effectiveness of the model approximation algorithm.

A behavioral approach to balanced representations of dynamical systems

The behavioral approach to linear systems provides an alternative framework for studying the notion of balanced representations. A new definition for balanced representations is proposed that is one-to-one related to a set of system invariants that is obtained by assuming a specific Hilbert space structure on the system behavior. This notion of balancing is more general than the prevailing notion of balancing in that it is well defined for nonstable systems and is independent of a particular (input-output) representation of the system. It is shown that Lyapunov, LQ̧G, and H ∞ balanced representations are obtained as a special case. An application for the problem of model approximation is discussed.

A note on reduced-order controller synthesis

It is proposed that reduced-order controllers for single-input single-output plants may be synthesized by first calculating a controller for some, not necessarily accurate, reduced-order model of the plant, and then refining the controller so as to account for the full plant. The controller refinement is formulated in terms of a model-approximation problem within a feedback loop. Closed-loop stability is then guaranteed provided that the approximation error is below a certain bound. >

The radial distribution function for a fluid of hard spheres at high densities

A radial distribution function for hard spheres is obtained, which improves at high densities on the Percus-Yevick function. This is achieved by solving the mean spherical integral equation for a Yukawa form, adjusted to satisfy the Carnahan-Starling equation of state. The solution to the related problem of the mean spherical model approximation for a model fluid which interacts via the hard sphere potential plus the Yukawa potential is reported.