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.     

Controller design for distributed systems based on Hankel-norm approximations

A procedure for obtaining a finite-dimensional reduced-order model for a class of distributed systems is developed. It is based on a combination of a modal approximation and an optimal Hankel-norm approximation and the approximation error can be accurately predicted. The reduced-order model can be used to design a precompensator for the original system as its performance and robust stability characteristics can be deduced from the reduced-order model and finitely many modal parameters. The procedure is applied to a parabolic system.     

Stable partial Padé approximations for reduced-order trnsfer functions

While the Padé approximation can often be used to produce good reduced-order transfer functions, the method suffers from the disadvantage that a stable reduced order-model is not always obtained for an original model that is stable. A method is described for obtaining stable partial Padé approximation reduced-order models. The method permits the determination of the effect of changes in the coefficients in the partial Padé approximation on the poles of the reduced-order transfer function by means of standard root locus plots.     

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

In 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.     

Hankel-norm model approximation for LPV systems with parameter-varying time delays

This article is concerned with Hankel-norm model approximation (HNMA) for linear parameter-varying systems with parameter-varying time delays. For a given stable system, our attention is focussed on the construction of reduced-order models, which approximate the original system well in a Hankel-norm sense. By applying the slack matrix approach, a delay-dependent sufficient condition is proposed for the robustly asymptotic stability with a Hankel-norm error performance for the error system. Then, the HNMA problem is solved by using the projection approach, which casts the model approximation into a sequential minimisation problem subject to linear matrix inequality constraints by employing the cone complementary linearisation algorithm. Finally, a numerical example is provided to illustrate the effectiveness of the proposed methods.     

Order reduction of SISO nonminimum-phase stochastic systems

We consider the order reduction problem for single-input/ single-output nonminimum-phase linear stochastic systems. The problem is formulated as that of approximating the original system-output process by another stochastic process generated via a reduced-order model. It is shown that in order to obtain a "correct" phase approximation, it is necessary to consider some higher order statistics of the original process in addition to the usual second-order statistics. A computationally feasible approximation criterion is proposed and analyzed. An example is also presented to illustrate the proposed method.     

Model reduction based on regional pole and covariance equivalentrealizations

In this paper a novel model reduction problem is studied for linear continuous-time time-invariant stochastic systems. The purpose of this problem is to design the reduced-order model so that it has the same dominant pole region and steady state output covariance as those of the original full-order model. The resulting reduced-order model can approximate the corresponding original full-order model in two important aspects, i.e., transient and steady state performances. Necessary and sufficient conditions for the existence of desired reduced-order models are established, and an explicit expression for these reduced order model is also presented. An illustrative example is used to demonstrate the effectiveness of the proposed design method     

Application of generalized block-pulse operational matrices for the approximation of continuous-time systems

The central theme of this paper is to apply generalized block-pulse operational matrices to approximate continuous-time systems. Generalized block-pulse operational matrices and the Routh approximation method are used together to find a low-order transfer function to approximate the original high-order transfer function. The Routh approximation method is used to preserve the stability of the original system by first determining the denominator coefficients of the reduced-order system. Generalized block-pulse operational matrices are then applied to determine numerator coefficients of the reduced-order system by optimally matching the unit step responses of the original and reduced-order systems. This new constrained time-domain matching approach not only yields more satisfactory results than previous methods, but also provides a more straightforward and efficient method for the approximation of continuous-time systems.     

Arnoldi-based model reduction for fractional order linear systems

In this paper, the Arnoldi-based model reduction methods are employed to fractional order linear time-invariant systems. The resulting model has a smaller dimension, while its fractional order is the same as that of the original system. The error and stability of the reduced model are discussed. And to overcome the local convergence of Padé approximation, the multi-point Arnoldi algorithm, which can recursively generate a reduced-order orthonormal basis from the corresponding Krylov subspace, is used. Numerical examples are given to illustrate the accuracy and efficiency of the proposed methods.     

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.     

An approximate-predictor approach to reduced-order models andcontrollers for distributed-parameter systems

Two reduced-order digital controllers for distributed parameter systems (DPS) are described. Reduced-order models approximate the optimal finite past predictor and error covariance for the full system to minimize an approximation to the Kullback-Leibler information distance (KLID). An LQG controller based on a reduced-order-system model is described. A reduced-order controller is found to minimize the KLID between the closed-loop system outputs with the full- and reduced-order controllers. Noncollocated control of a flexible beam is simulated     

离散随机模型降阶的最小条件信息损失方法

针对随机系统的模型降阶问题,从分析离散线性随机状态方程模型中的条件信息描述机制入手,讨论了模型状态集聚过程中系统的平均条件信息损失.运用在模式识别领域中获得成功应用的最小信息损失准则得出了一种新的模型降阶信息论方法———基于状态集聚的最小条件信息损失方法,并讨论了降阶模型阶次的选择.分析表明,当原系统是渐近稳定时,由该方法得出的降阶模型也是渐近稳定的.该方法运用简单,仿真研究也表明由该方法得出的降阶模型具有良好的近似性能.     

Frequency weighted model reduction via optimal projection

The problem of order reduction with frequency weighting is considered. Necessary conditions characterizing the reduced-order model which minimizes a quadratic error criterion in the frequency domain, with either input or output frequency weighting, are given. The problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input. When the weighting matrix is singular at infinity, the reduced-order model has a direct transmission term even if the original system is strictly proper. This additional degree of freedom can be used to obtain better approximation over a prespecified frequency range     

Reduced-order modelling of linear MIMO systems with the Padé approximation method

The Padé approximation method, coupled with three powerful dominant pole selection criteria, is introduced for the purpose of application to the high-degree matrix transfer function (which may be derived by the application of the Leverrier algorithm to the MIMO linear time-invariant state-space representation) of the original system and obtaining corresponding adequate reduced-order model(s). The proposed method has been applied successfully to the matrix function (obtained by the application of the Leverrier algorithm) of a lOth-order two-input two-output time-invariant linear model of a practical power system.     

H ∞ model reduction for discrete time-delay systems: delay-independent and dependent approaches

This paper investigates the problem of H _∞ model reduction for linear discrete-time state-delay systems. For a given stable system, our attention is focused on the construction of reduced-order models, which guarantee the corresponding error system to be asymptotically stable and have a prescribed H _∞ error performance. Both delay-independent and dependent approaches are developed, with sufficient conditions obtained for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearization method is exploited to cast them into sequential minimization problems subject to LMI constraints, which can be readily solved in standard numerical software. In addition, the development of reduced-order models with special structures, such as delay-free models and zeroth-order models, is also addressed. The approximation methods presented in this paper can be further extended to cope with systems with uncertain parameters. Two numerical examples have been provided to show the effectiveness of the proposed theories.     

Reduced-order models and controllers for continuous-time stochasticsystems: an information theory approach

Reduced-order models and controllers for continuous-time stochastic systems are described. The reduced-order models are chosen to minimize the Kullback-Leibler information distance (KLID) between the outputs of the actual and reduced systems. An LQG controller based on a reduced-order system model is described. A second reduced-order controller is found to minimize the KLID between the closed-loop system outputs with the full and reduced-order controllers     

Comments on the computation of interval Routh approximants

In Bandyopadhyay et al. (1994, 1997), the Routh approximation method was extended to derive reduced-order interval models for linear interval systems. In this paper, the authors show that: 1) interval Routh approximants to a high-order interval transfer function depend on the implementation of interval Routh expansion and inversion algorithms; 2) interval Routh expansion algorithms cannot guarantee the success in generating a full interval Routh array; 3) some interval Routh approximants may not be robustly stable even if the original interval system is robustly stable; and 4) an interval Routh approximant is in general not useful for robust controller design because its dynamic uncertainties (in terms of robust frequency responses) do not cover those of the original interval system     

A realization approach to stochastic model reduction

The problem of discrete-time stochastic model reduction (approximation) is considered. Using the canonical correlation analysis approach of Akaike (1975), a new order-reduction algorithm is developed. Furthermore, it is shown that the inverse of the reduced-order realization is asymptotically stable. Next, an explicit relationship between canonical variables and the linear least-squares estimate of the state vector is established. Using this, a more direct approach for order reduction is presented, and also a new design for reduced-order Kalman filters is developed. Finally, the uniqueness and symmetry properties for the new realization—the balanced stochastic realization—along with a simulation result, are presented.     

L1 adaptive control for general partial differential equation (PDE) systems

This paper addresses the L₁ adaptive control problem for general Partial Differential Equation (PDE) systems. Since direct computation and analysis on PDE systems are difficult and time-consuming, it is preferred to transform the PDE systems into Ordinary Differential Equation (ODE) systems. In this paper, a polynomial interpolation approximation method is utilized to formulate the infinite dimensional PDE as a high-order ODE first. To further reduce its dimension, an eigenvalue-based technique is employed to derive a system of low-order ODEs, which is incorporated with unmodeled dynamics described as bounded-input, bounded-output (BIBO) stable. To establish the equivalence with original PDE, the reduced-order ODE system is augmented with nonlinear time-varying uncertainties. On the basis of the reduced-order ODE system, a dynamic state predictor consisting of a linear system plus adaptive estimated parameters is developed. An adaptive law will update uncertainty estimates such that the estimation error between predicted state and real state is driven to zero at each time-step. And a control law is designed for uncertainty handling and good tracking delivery. Simulation results demonstrate the effectiveness of the proposed modeling and control framework.     

Loop细分曲面的优化拟合算法

提出一种用于构造给定三维模型的拟合Loop细分曲面的迭代优化算法,使得拟合曲面与原始模型之间的逼近误差最小.算法中的逼近误差定义为原始模型各面元到拟合曲面最小距离的积分.与Loop细分小波分解算法的比较表明,该算法以适度的运行时间代价得到了更优的结果.此外,该算法还可以加以推广,作为一类从输入模型生成其近似表示的优化算法的基础.