Aircraft turbofan engine linear model with uncertain eigenvalues

This paper presents an aircraft turbofan engine model identification in time domain. The identification makes use of a multivariable linear model of turbofan engine dynamics having uncertain eigenvalues. The uncertainty describes the difference between nonlinear and linear models. The bounds for real and imaginary parts of uncertain matrix eigenvalues are determined using nonlinear programming. The results of identification application using a detailed nonlinear model of an aircraft turbofan engine are presented.     

Fractional balanced reduction: model reduction via fractionalrepresentation

A method for the model reduction of finite-dimensional, linear, time-invariant (FDLTI) plants is proposed which uses normalized fractional representations is proposed. The method, dubbed fractional balanced reduction, applies balance and truncate to a special representation of the graph operator of the plant. This operation yields the graph operator of a reduced order plant. The method has such properties as existence of an a priori error bound in the graph metric and preservation of sector containment. Coupling fractional representations with principal component analysis gives a model reduction method that is capable of producing, in a numerically stable way, a good reduced order model using the whole full order model. Sector properties are also preserved-these are useful for deciding stability when nonlinearities are in the loop     

Parameter independent model order reduction

Several recently developed model order reduction methods for fast simulation of large-scale dynamical systems with two or more parameters are reviewed. Besides, an alternative approach for linear parameter system model reduction as well as a more efficient method for nonlinear parameter system model reduction are proposed in this paper. Comparison between different methods from theoretical elegancy to complexity of implementation are given. By these methods, a large dimensional system with parameters can be reduced to a smaller dimensional parameter system that can approximate the original large sized system to a certain degree for all the parameters.     

Internal positivity preserved model reduction

This article studies model reduction of continuous-time stable positive linear systems under the Hankel norm, H _∞ norm and H ₂ norm performance. The reduced-order systems preserve the stability as well as the positivity of the original systems. This is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the Lyapunov matrices are decoupled with the system matrices. In this way, the positivity constraints in the reduced-order model can be imposed in a natural way. As the model reduction performances are expressed in linear matrix inequalities with equality constraints, the desired reduced-order positive models can be obtained by using the cone complementarity linearisation iterative algorithm. A numerical example is presented to illustrate the effectiveness of the given methods.     

Remarks on stochastic model reduction

In this note, we give a solution to the stochastic system model reduction problem in terms of the approximation of the output stochaslic process power spectrum by applying balanced realization or optimal Hankel norm approximation to the asymptotically stable and minimum phase spectral factor. We also give explicit bounds for the approximation error. The effectiveness of this method is compared to the one by Jonckheere and Helton.     

Structural equivalence and model reduction

Within the framework of structural analysis, techniques are developed for obtaining reduced-order structural models in which all structural information is preserved. The reduced models are valid for any input and output sets guaranteeing that, from the structural controllability test, the same set of feasible input-output pairings are obtained from the low-order model as would be obtained from the high-order model. New insight into the practical significance of the generic rank test for structural controllability is provided, and an algorithm for the development of low-order models for staged systems is presented.     

Time-interval model reduction of bilinear systems

In this paper, a new method for model reduction of bilinear systems is presented. The proposed technique is from the family of gramian-based model reduction methods. The method uses time-interval generalised gramians in the reduction procedure rather than the ordinary generalised gramians and in such a way that it improves the accuracy of the approximation within the time-interval for which the method is applied. The time-interval generalised gramians are the solutions to the generalised time-interval Lyapunov equations. The conditions for these equations to be solvable are derived and an algorithm is proposed to solve these equations iteratively. The method is further illustrated with the help of two illustrative examples. The numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians.     

A modified stochastic perturbation algorithm for closely-spaced eigenvalues problems based on surrogate model

Aiming at uncertainty propagation and dynamic reanalysis of closely-spaced eigenvalues, with consideration of uncertainties in design variables, a modified stochastic perturbation method is proposed. Concerning quasi-symmetric or partial-symmetric structures that frequently appear, one of their primary features is closely-distributed natural frequencies. For structure with closely-spaced eigenvalues, due to its instability and sensitivity to the changes of design variables and its excessively concentrated adjacent eigenvalues, conventional uncertainty analysis or dynamic reanalysis methods for distinct eigenvalue are no longer available. Initially, the spectral decompositions of stiffness and mass matrices are provided; by transfer technique, the eigen-problem of closely-spaced eigenvalues is converted to that of repeated eigenvalues with two perturbation parts appended; then the perturbed closely-spaced eigenvalue is rewritten as the sum of original closely-spaced eigenvalues’ mean value and surrogate model which approximates the first-order perturbation term by polynomial chaos expansions. According to this method, statistical quantities of perturbed closely-spaced eigenvalues are calculated directly and accurately, which contributes to its uncertainty analysis and dynamic reanalysis. Furthermore, the capability of proposed method in dealing with relatively large uncertainties and complex engineering structure is demonstrated. The accuracy and efficiency of proposed method have been verified sufficiently by numerical examples.     

Abstract reduction in directed model checking CCS processes

Model checking tools face a combinatorial blow up of the state-space (commonly known as the state explosion problem) that must be addressed to formally verify concurrent systems. We propose an approach combining abstraction techniques and heuristic search to overcome the problem above. In particular, heuristic search can avoid the bottleneck of the exhaustive exploration of the global state graph of a system, while retaining the advantages of abstraction techniques.     

Model reduction in the presence of uncertainty in model parameters

This short note describes how to extend a certain class of existing model reduction techniques to take into account uncertainty in model parameters. The key idea of this extension is that the reduced-order model should not only contain the model parameters, but that the reduction procedure itself has to be geared for dealing with parametric uncertainty. This goal is achieved by augmenting the vector of inputs to the system with the uncertain parameters and by performing model reduction on the augmented system. It is shown that error bounds for the reduced-order model can be computed if the underlying system is linear with respect to the states, parameters, and inputs. A comparison between the presented technique and a conventional approach is made via two examples.     

模型的快速降阶技术

为快速进行模型的降阶,结合平衡截断（Balanced Truncation,BT）方法和特征正交分解（Proper Orthogonal Decomposition,POD）方法提出一种模型降阶方法.该方法采用频域POD快照矩阵低阶逼近系统的可控、可观Gram矩阵;通过奇异值分解（Singular Value Decomposition,SVD）提取BT＋POD模态,对低能量模态截断形成降阶子空间,并将其映射到全阶系统,从而形成基于状态空间的降阶模型（Reduced Order Model,ROM）;该模型就成为全阶模型（Full Order Model,FOM）的ROM.通过对阶数n=406的LTI SISO系统和阶数n=9的2区间电力系统进行的验证表明,在保留BT方法输入输出平衡特性的基础上,该方法效率高于BT方法.     

Deterministic balancing and stochastic model reduction

A recent approach to the deterministic model reduction problem is based on the notion of balancing. However, the original development of deterministic balancing did not contain any statistical considerations with which to develop a stochastic model reduction algorithm. Nevertheless, it is shown in this note that there are two stochastic model reduction algorithms in the literature which result in a deterministically balanced model. Their equivalence with deterministic balancing provides a stochastic interpretation to the deterministic algorithm.     

A frequency domain model reduction procedure

This report presents a method for obtaining low order models of high order stable linear systems which match the frequency response of the high order system at a set of prespecified frequencies. The model is obtained by measuring filtered input-output data generated by a single periodic test input. The method is applicable to both scalar and multivariable systems, and it is illustrated by application to a 16th order model of the F-100 turbofan engine.     

On Routh-Pade model reduction of interval systems

This paper discusses another generalization of the direct Routh table truncation method for interval systems. It is shown that the existing generalization of the direct Routh table truncation fails to produce a stable system, in contradiction to the equivalent result for fixed-coefficients systems. The present method guarantees a stable reduced order model for interval systems as well.     

无人机侧向系统LPV 模型降阶

无人机线性参变(LPV)模型能准确描述其非线性动态特性,但初始建立的LPV模型阶数较高,控制过程计算量较大.为此,提出一种基于平衡截断的LPV模型降阶方法.首先给出LPV系统的适定性、稳定性和平衡实现的定义;然后,提出LPV模型的平衡截断降阶方法.针对无人机侧向系统LTI模型,通过多项式拟合来建立LPV模型,并实现模型降阶.仿真结果表明,降阶模型的阶跃响应满足输出响应的精度要求.     

Multiple eigenvalues in structural optimization problems

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.Dedicated to the memory of Ernest F. MasurGuest professor during the period 16 November to 11 December, 1992 and 15 November to 12 December, 1993.     

On model reduction of K-power bilinear systems

As a special type of bilinear systems, K-power bilinear systems have a special coupled structure that should be preserved in the process of model reduction. We investigate moment matching methods for K-power systems and extract structure-preserved reduced models from the perspective of bilinear systems and coupled systems. The optimal H₂ reduction is also considered for K-power systems. We prove that there exist reduced models satisfying the optimality conditions and meanwhile preserving the coupled structure of the original models. Furthermore, such reduced models can be produced by an iterative algorithm, or alternatively by a subsystem-iteration algorithm with less computational effort and faster convergence rate. Simulation results show that the proposed iterative algorithms possess superior performance in contrast to moment matching methods.     

On model reduction of discrete time systems

In this paper, further justification of model reduction by Badreddin-Mansour's method is presented. In particular, connections between polynomial root-location properties, coefficient properties and Schur-Cohn coefficient properties are established. Lattice realizations of a discrete system and its reduced model are given, which illustrate certain relationships between the original model and its reduced form. These relationships give insight into the validity of the method of reduction and offer insight on the order of approximation.