An Improved Parameterized Controller Reduction Technique via New Frequency Weighted Model Reduction Formulation

In this paper, an improved parameterized controller reduction technique via a new frequency weighted model reduction formulation is developed for the feedback control of MIMO discrete time systems particularly for non‐unity feedback control system configurations which have the controller located in the feedback path. New frequency weights which are a function of a free parameter matrix are derived based on a set of equivalent block diagrams and this leads to a generalized double sided frequency weighted model reduction formulation. Solving this generalized double sided frequency weighted model reduction problem for various values of the free parameter results in obtaining controllers which correspond to each value of the free parameter. It is shown that the proposed formulation has a useful characteristic such that selecting a controller which corresponds to a large value of the free parameter results in obtaining an optimal reduced order controller and using this optimal reduced order controller in a closed loop system results in significant reduction in the infinity norm of the approximation error between the original closed loop system and the closed loop system which uses an optimal reduced order controller (when compared to existing frequency weighted model reduction methods).     

Partial Order Reduction for Verification of Spatial Properties of Pi-Calculus Processes

Mechanical tools have recently been developed that enable computer-aided verification of spatial properties of concurrent systems. To be practical, these tools are expected to deal with the state- space explosion problem. In order to alleviate this problem, we develop partial order reduction for verification of spatial properties of pi-calculus processes. The main issue is that spatial logics are very expressive and some spatial formulas prevent partial order reduction. After discussing this issue, we propose a restricted spatial logic such that partial order reduction holds. Our approach relies on exploiting partially confluent communications and on identifying invisible communications in the pi-calculus, for which we propose a simple syntactic criterion.     

Generalised gramian framework for model/controller order reduction of switched systems

In this article, a general method for model/controller order reduction of switched linear dynamical systems is presented. The proposed technique is based on the generalised gramian framework for model reduction. It is shown that different classical reduction methods can be developed into a generalised gramian framework. Balanced reduction within a specified frequency bound is developed within this framework. In order to avoid numerical instability and also to increase the numerical efficiency, generalised gramian‐based Petrov–Galerkin projection is constructed instead of the similarity transform approach for reduction. The framework is developed for switched controller reduction. To the best of our knowledge, there is no other reported result on switched controller reduction in the literature. The method preserves the stability under an arbitrary switching signal for both model and controller reduction. Furthermore, it is applicable to both continuous and discrete time systems for different classical gramian‐based reduction methods. The performance of the proposed method is illustrated by numerical examples.     

De Bruijn''s syntax and reductional behaviour of λ-terms: the typed case

In this paper, we consider the typed versions of the λ-calculus written in a notation which helps describe canonical forms more elegantly than the classical notation, and enables to divide terms into classes according to their reductional behaviour. In this notation, β-reduction can be generalised from a relation on terms to one on equivalence classes. This class reduction covers many known notions of generalised reduction. We extend the Barendregt cube with our class reduction and show that the subject reduction property fails but that this is not unique to our class reduction. We show that other generalisations of reduction (such as the σ-reduction of Regnier) also behave badly in typed versions of the λ-calculus. Nevertheless, solution is at hand for these generalised reductions by adopting the useful addition of definitions in the contexts of type derivations. We show that adding such definitions enables the extensions of type systems with class reduction and σ-reduction to satisfy all the desirable properties of type systems, including subject reduction and strong normalisation. Our proposed typing relation ^c is the most general relation in the literature that satisfies all the desirable properties of type systems. We show that classes contain all the desirable information related to a term with respect to typing, strong normalisation, subject reduction, etc.     

相容关系下不协调集值目标信息系统知识约简

在不协调集值目标信息系统中定义了相容关系,并在基于相容关系的不协调集值目标信息系统中引入分配约简和近似约简的概念,讨论了二者之间的关系,并且得到了分配约简和近似约简的判定定理以及辨识矩阵,建立了不协调集值目标信息系统的分配和近似约简的具体方法,同时通过实例验证了该方法的有效性。     

Order reduction for nonlinear dynamic models of distributed reacting systems

Detailed first-principles models of transport and reaction (based on partial differential equations) lead, after discretization, to dynamical systems of very high order. Systematic methodologies for model order reduction are vital in exploiting such fundamental models in the analysis, design and real-time control of distributed reacting systems. We briefly review some approaches to model order reduction we have successfully used in recent years, and illustrate their capabilities through (a) the design of an observer and stabilizing controller of a reaction-diffusion problem and (b) two-dimensional simulations of the transient behavior of a horizontal MOVPE reactor.     

控制系统最优降阶技术与应用

本文给出一种新的控制系统最优降阶技术并介绍其在控制系统设计中的应用，文中引入的最优降阶的准则是在指定输入信号时，对降阶模型与原始模型之间误差的加权最小积分方差。广西重点讨论连续系统的降阶算法，但结论同样适合于离散时间系统，文中还将通过一个例子来淙本方法和其它传统方法相比的优越性。     

不协调区间值决策系统的分布约简

知识约简可以保持决策系统中的分类特征不变,是粗糙集理论的重要研究内容之一。分布约简保持约简前后决策系统中各规则的置信度不发生改变。为了给区间值决策系统的论域分类提供合理的度量标准,引入了区间值相似率。通过将Pawlak决策系统中的等价关系扩展到区间值决策系统中的相容关系,提出了区间值决策系统的分布约简目标。针对该目标给出了相应差别矩阵的计算方法,并与现有区间值决策系统的广义决策约简计算方法进行了分析比较。最后,通过人工数据集的实验验证了相关结论的有效性。     

Distributed Partial Order Reduction of State Spaces

State space explosion is a fundamental obstacle in formal verification of concurrent systems. Several techniques for combating this problem have emerged in the past few years, among which the two we are interested in are: partial order reduction and distributed memory state exploration. While the first one tries to reduce the problem to a smaller one, the other one tries to extend the computational power to solve the same problem. In this paper, we consider a combination of these two approaches and propose a distributed memory algorithm for partial order reduction.     

Multi-order Arnoldi-based model order reduction of second-order time-delay systems

In this paper, we discuss the Krylov subspace-based model order reduction methods of second-order systems with time delays, and present two structure-preserving methods for model order reduction of these second-order systems, which avoid to convert the second-order systems into first-order ones. One method is based on a Krylov subspace by using the Taylor series expansion, the other method is based on the Laguerre series expansion. These two methods are used in the multi-order Arnoldi algorithm to construct the projection matrices. The resulting reduced models can not only preserve the structure of the original systems, but also can match a certain number of approximate moments or Laguerre expansion coefficients. The effectiveness of the proposed methods is demonstrated by two numerical examples.     

On H2 model reduction of bilinear systems

The H₂ model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H₂ norm of bilinear systems in terms of the state-space matrices, the H₂ model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H₂ optimal are given. The gradient flow approach is used to obtain the solution of the H₂ model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.     

A Global Maximum Error Controller Based Nonlinearity Aware TPWL Method for Reducing Nonlinear Systems

Abstract

Model order reduction is a common practice to reduce large order systems so that their simulation and control become easy. Nonlinearity aware trajectory piecewise linear is a variation of trajectory piecewise linearization technique of order reduction that is used to reduce nonlinear systems. With this scheme, the reduced approximation of the system is generated by weighted sum of the linearized and reduced sub-models obtained at certain linearization points on the system trajectory. This scheme uses dynamically inspired weight assignment that makes the approximation nonlinearity aware. Just as weight assignment, the process of linearization points selection is also important for generating faithful approximations. This article uses a global maximum error controller based linearization points selection scheme according to which a state is chosen as a linearization point if the error between a current reduced model and the full order nonlinear system reaches a maximum value. A combination that not only selects linearization points based on an error controller but also assigns dynamic inspired weights is shown in this article. The proposed scheme generates approximations with higher accuracies. This is demonstrated by applying the proposed method to some benchmark nonlinear circuits including RC ladder network and inverter chain circuit and comparing the results with the conventional schemes.     

Reduction of stable differential–algebraic equation systems via projections and system identification

Most large-scale process models derived from first principles are represented by nonlinear differential–algebraic equation (DAE) systems. Since such models are often computationally too expensive for real-time control, techniques for model reduction of these systems need to be investigated. However, models of DAE type have received little attention in the literature on nonlinear model reduction. In order to address this, a new technique for reducing nonlinear DAE systems is presented in this work. This method reduces the order of the differential equations as well as the number and complexity of the algebraic equations. Additionally, the algebraic equations of the resulting system can be replaced by an explicit expression for the algebraic variables such as a feedforward neural network. This last property is important insofar as the reduced model does not require a DAE solver for its solution but system trajectories can instead be computed with regular ODE solvers. This technique is illustrated with a case study where responses of several different reduced-order models of a distillation column with 32 differential equations and 32 algebraic equations are compared.     

Model reduction via time-interval balanced stochastic truncation for linear time invariant systems

In this article, a new method for model reduction of linear dynamical systems is presented. The proposed technique is from the family of gramian-based relative error model reduction methods. The method uses time-interval gramians in the reduction procedure rather than ordinary gramians and in such a way it improves the accuracy of the approximation within the time interval which is applied. It is proven that the reduced order model is stable when the proposed method applies to a stable system. The method uses a recently proposed inner–outer factorisation algorithm which enhances the numerical accuracy and efficiency. In order to avoid numerical instability and also to further increase the numerical efficiency, projector matrices are constructed instead of the similarity transform approach for reduction. The method is illustrated by a numerical example and finally it is applied to a practical CD player example. The numerical results show that the method is more accurate than ordinary balanced stochastic truncation.     

Relaxed Visibility Enhances Partial Order Reduction

State-space explosion is a central problem in the automatic verification (model-checking) of concurrent systems. Partial order reduction is a method that was developed to try to cope with the state-space explosion. Based on the observation that the order of execution of concurrent (independent) atomic actions is in many cases unimportant for the checked property, it allows reducing the state space by exploring fewer execution sequences. However, in order to guarantee that the reduced state space preserves the correctness of the checked property, the partial order reductions put constraints about commuting the order of atomic actions that may change the value of propositions appearing in the checked specification. In this paper we relax this constraint, allowing a weaker requirement to be imposed, and thus achieving a better reduction. We demonstrate the benefits of our improved reduction with experimental results.     

A novel order reduction method for nonlinear dynamical system under external periodic excitations

The concept of approximate inertial manifold (AIM) is extended to develop a kind of nonlinear order reduction technique for non-autonomous nonlinear systems in second-order form in this paper.Using the modal transformation,a large nonlinear dynamical system is split into a 'master' subsystem,a 'slave' subsystem,and a 'negligible' subsystem.Accordingly,a novel order reduction method (Method I) is developed to construct a low order subsystem by neglecting the 'negligible' subsystem and slaving the 'slave' sub...     

Parameterization theory of balanced truncation method for any evenly distributed RC interconnect circuits and transmission lines

This paper presents a parameterization theory of balanced truncation method (BTM) for any evenly distributed RC interconnect and transmission lines and their BTM reduction models. The parameterization theory shows that any evenly distributed RC line circuits of the same order have the same balanced gramian, BTM error upper bound, and BTM approximation error, which are independent of their RC parameters. Thus, the prototype model is proposed for research to dramatically reduce the computations for various RC parameters. Under normalized (scaled) time axis and frequency axis, any time responses and the Bode plots of their BTM reduction models are also the same as the ones of the prototype BTM reduction models, respectively. The simulations demonstrate the theory. The reduction model order can be identified by the prototype model. The new results can be applied to not only the model reduction of interconnects or transmission lines, but also control systems with transmission lines, networks, or time delay units.     

Continuous time LTI systems under lossless positive real transformations: open-loop balanced representation and truncated reduced-order models

In this paper, new results on the open-loop balanced representation of continuous time linear time-invariant systems are reported. More particularly, the effect of lossless positive real transformations on open-loop balanced representations is investigated with specific attention to the problem of model order reduction. The properties of systems where a lossless positive real transformation has been applied are discussed showing that, if the original system is open-loop balanced, the resulting transformed system is still open-loop balanced. Furthermore, the singular values of the transformed system are related to those of the original one. These results allow to derive a model order reduction strategy for this class of systems that leads to a consistent decrease of the numerical complexity. The proposed approach reveals to be of particular interest for the design of reduced-order systems with specific amplitude responses, including analog multiband filters.     

属性序下基于优势关系灰色信息系统约简

粗糙集和灰色理论在数据挖掘领域各有优点,它们最终目标都是为了发现知识。将粗糙集和灰色系统结合,研究了区间灰色信息系统在优势关系下的约简方法。在现实世界中,属性之间可能存在着一种属性序的关系,将这种属性序关系引入到基于优势关系的灰色信息系统中,给出了两种属性约简的算法：一种基于分辨矩阵的算法,另一种不基于分辨矩阵的算法。较好地适应了不完备信息系统和大数据集信息系统约简。最后通过实例对比,得出两种算法的约简结果是一致的,并分析了各自的适用范围。     

不协调目标信息系统的知识约简

在不协调目标信息系统中引入了最大分布约简的概念，讨论了最大分布约简、分配约简、分布约简和近拟约简之间的关系。最大分布 间弱于分布约简，克服了对信息系统过于苛刻的要求。同时，它又克服了分配约简可能产生与原系统不相容的命题规则的缺陷；给出了这些知识约简的判定定理和相应的可辨识属性矩阵，从而提供了不协调目标信息系统的知识约简的新方法。