Cheap computation of optimal reduced-order models for systems with time delay

A simple method of optimally reducing the order of systems with time delays is proposed. The performance indices used for optimization are the integrals of the time-weighted squared error between the responses of the reduced-order and original models. The performance indices are first expressed in terms of the reduced order system unknown parameters and a minimization of these indices gives the reduced model optimal parameters. Cheap and accurate computation of optimal reduced order models with time delay has so far proved abortive as closed-form expressions for these indices have proved difficult to obtain. Simple feedback controllers parametrized by the internal model control method using the reduced models result in excellent performance when implemented on the original model.     

离散系统区域极点/协方差等价实现一种代数方法

提出了离散随机系统的模型简化新方法:区域极点/协方差等价实现方法.即构造降阶模型,使其匹配给定的区域极点和稳态协方差值,在暂态性能和稳态性能这两个重要方面近似于给定的满阶模型.文中证明了满足要求的降阶模型的存在性,并直接给出了降阶模型的解析表达式,最后提供了说明性的数值例子.     

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.     

Assessment of stability robustness for adaptive controllers

The problem of improving the stability characteristics of discrete-time SISO reduced order adaptive controllers is addressed in this note. A method is proposed to adjust the observer poles of an indirect adaptive scheme in order to ensure stability of the closed-loop system in spite of possible unstructured uncertainties (e.g., process-model order mismatch). Assuming that the identified model transfer function module converges within a previously defined uncertainty range, the frequency response-like compensation method is shown to lead to a stable design, provided that the reduced order estimated model is stable and stably invertible and the system is open-loop stable.     

Identification of multi-input systems: variance analysis and input design issues

This paper examines the identification of multi-input systems. Motivated by an experiment design problem (should one excite the various inputs simultaneously or separately), we examine the effect of an additional input on the variance of the estimated coefficients of parametrized rational transfer function models, with special emphasis on the commonly used FIR, ARX, ARMAX, OE and BJ model structures. We first show that, for model structures that have common parameters in the input-output and noise models (e.g. ARMAX), any additional input contributes to a reduction of the covariance of all parameter estimates. We then show that the accuracy improvement extends beyond the case of common parameters in all transfer functions, and we show exactly which parameter estimates are improved when a new input is added. We also conclude that it is always better to excite all inputs simultaneously.     

H∞ model reduction of Markovian jump linear systems

In this paper, the H_∞ model reduction problem for linear systems that possess randomly jumping parameters is studied. The development includes both the continuous and discrete cases. It is shown that the reduced order models exist if a set of matrix inequalities is feasible. An effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions. Using the numerical solutions of the matrix inequalities, the reduced order models can be obtained. An example is given to illustrate the proposed model reduction method.     

Model order reduction of MIMO bilinear systems by multi-order Arnoldi method

In this paper, we present a time domain model order reduction method for multi-input multi-output (MIMO) bilinear systems by general orthogonal polynomials. The proposed method is based on a multi-order Arnoldi algorithm applied to construct the projection matrix. The resulting reduced model can match a desired number of expansion coefficient terms of the original system. The approximate error estimate of the reduced model is given. And we also briefly discuss the stability preservation of the reduced model in some cases. Additionally, in combination with Krylov subspace methods, we propose a two-sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously. The effectiveness of the proposed methods is demonstrated by two numerical examples.     

A high-performance parallel implementation of the certified reduced basis method

The certified reduced basis method (herein RB method) is a popular approach for model reduction of parametrized partial differential equations. In this paper we introduce new techniques that are required to efficiently implement the Offline “Construction stage” of the RB method on high-performance parallel supercomputers. This enables us to generate certified RB models for large-scale three-dimensional problems that can be evaluated on standard workstations and other “thin” computing resources with speedup of many orders of magnitude compared to the corresponding full order model. We use our implementation to perform detailed numerical studies for two computationally expensive model problems: a natural convection fluid flow problem and a “many parameter” heat transfer problem. In the heat transfer problem, we exploit the computational efficiency of the RB method to perform a detailed study of “snapshot” selection in the Greedy algorithm, and we also examine statistics of the output sensitivity derivatives to obtain a “global” view of the relative importance of the parameters.     

A frequency weighted model order reduction technique and error bounds

A new frequency weighted technique for balanced model reduction is proposed. The proposed technique not only provides stable reduced order models for the case when both input and output weightings are included but also yields frequency response error bounds. The method is illustrated using numerical examples and the results are compared with other frequency weighted model reduction techniques.     

H∞ identification of “soft” uncertainty models

The paper investigates the problem of identifying uncertainty models of causal, SISO, LTI, discrete-time, BIBO stable, unknown systems, using frequency domain measurements corrupted by Gaussian noise of known covariance. Additive uncertainty models are looked for, consisting of a nominal model and an additive dynamic perturbation accounting for the modeling error. The nominal model is chosen within a class of affinely parametrized models with transfer function of given (possibly low) order. An estimate of the parameters minimizing the H_∞ modeling error is obtained by minimizing an upper bound of the worst-case (with respect to the modeling error) second moment of the estimation error. Then, a bound in the frequency domain guaranteeing to include, with probability α, the frequency response error between the estimated nominal model and the unknown system is derived.     

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.     

Stabilization of sampled-data nonlinear systems by receding horizon control via discrete-time approximations

Results on stabilizing receding horizon control of sampled-data nonlinear systems via their approximate discrete-time models are presented. The proposed receding horizon control is based on the solution of Bolza-type optimal control problems for the parametrized family of approximate discrete-time models. This paper investigates both situations when the sampling period T is fixed and the integration parameter h used in obtaining approximate model can be chosen arbitrarily small, and when these two parameters coincide but they can be adjusted arbitrary. Sufficient conditions are established which guarantee that the controller that renders the origin to be asymptotically stable for the approximate model also stabilizes the exact discrete-time model for sufficiently small integration and/or sampling parameters.     

Estimation of an N-L-N Hammerstein-Wiener model

Estimation of a single-input single-output block-oriented model is studied. The model consists of a linear block embedded between two static nonlinear gains. Hence, it is called N-L-N Hammerstein-Wiener model. First, the model structure is motivated and the disturbance model is discussed. The paper then concentrates on parameter estimation. A relaxation iteration scheme is proposed by making use of a model structure in which the error is bilinear-in-parameters. This leads to a simple algorithm which minimizes the original loss function. The convergence and consistency of the algorithm are studied. In order to reduce the variance error, the obtained linear model is further reduced using frequency weighted model reduction. Simulation study will be used to illustrate the method.     

Improved component mode synthesis and variants

This survey focuses on the two known model order reduction schemes being widely integrated in various commercial finite element packages, namely, the static and dynamic condensation methods. The advantages as well as the corresponding drawbacks have been extensively analyzed in several papers throughout the last decades. Based on combining the beneficial properties of the aforementioned methods, several alternative reduction methodologies are outlined in this paper, i.e., the generalized improved reduction system method, the generalized component mode synthesis and the improved component mode synthesis with its generalized version, which incorporate in a more efficient way the system’s inertia terms. Therefore, the associated error regarding higher frequency ranges of interest is better controlled. Basis of these methodologies is the so-called master and slave degrees of freedom partitioning, the right selection of which highly influences the reduced order model’s dynamics. The methods are tested and verified on a rather small three-dimensional bar structure and on the lever part of a turbocharger’s variable turbine geometry. Several reduced order models are generated by varying both the number of Craig–Bampton modes and the selection of the required master degrees of freedom. A comparison is conducted based on the modal criterion of the corresponding eigenvectors and the associated computation time required.     

Identification of Hammerstein models for control using ASYM

Identification of single-input single-output Hammerstein models is studied in this work. The basic idea here is to extend the recently developed asymptotic method (ASYM) of linear model identification to include input non-linearity in the model set. First identification test design will be discussed. In parameter estimation, prediction error criterion is used in order to maintain consistence when the process is operating in closed-loop. A relaxation iteration scheme is proposed by making use of a model structure in which the error is bilinear in the parameters. The order of the linear part and nonlinear part are determined by looking at an output error related criterion which is control-relevant. The frequency domain upper error bound of the linear part will be derived and used for model validation. Simulation study will be used to illustrate the method and comparisons with other methods are also given.     

Frequency limited Gramians-based structure preserving model order reduction for discrete time second-order systems

A new technique for frequency limited model order reduction of discrete time second-order systems is presented. Discrete time frequency limited Gramians (DFLGs) and corresponding discrete algebraic Lyapunov equations are developed. An efficient technique for the computation of DFLGs and their Cholesky factors is presented. Computed DFLGs are partitioned to obtain position and velocity Gramians. These Gramians are balanced with different combinations to obtain various balanced transformations that yield Hankel singular values (HSVs) for order reduction. Frequency limited discrete time balanced truncation framework is proposed and truncation based on magnitudes of HSVs is applied to obtain the reduced order model. Moreover, stability conditions for reduced order models are stated. Results of the proposed technique are compared with infinite Gramians balancing scheme in order to certify the usefulness of the presented technique for frequency limited applications.     

PID auto-tuning using new model reduction method and explicit PID tuning rule for a fractional order plus time delay model

In this paper, a new model reduction method and an explicit PID tuning rule for the purpose of PID auto-tuning on the basis of a fractional order plus time delay model are proposed. The model reduction method directly fits the fractional order plus time delay model to frequency response data by solving a simple single-variable optimization problem. In addition, the optimal tuning parameters of the PID controller are obtained by solving the Integral of the Time weighted Absolute Error (ITAE) minimization problem and then, the proposed PID tuning rule in the form of an explicit formula is developed by fitting the parameters of the formula to the obtained optimal tuning parameters. The proposed tuning method provides almost the same performance as the optimal tuning parameters. Simulation study confirms that the auto-tuning strategy based on the proposed model reduction method and the PID tuning rule can successfully incorporate various types of process dynamics.     

Model order reduction of nonlinear models based on decoupled multi-model via trajectory piecewise linearization

In this paper a novel model order reduction method for nonlinear models, based on decoupled multi-model, via trajectory piecewise-linearization is proposed. Through different strategies in trajectory piecewise-linear model reduction, model order reduction of a trajectory piecewise-linear model based on output weighting (TPWLOW), has been developed by authors of current work. The structure of mentioned work was founded based on Krylov subspace method, which is appropriate for high order models. Indeed the size of the Krylov subspaces may increase with the number of inputs of the system. As a result, the use of Krylov subspace method may become deficient the case for multi-input systems of order few decades. This paper aims to generalize the idea of model reduction of TPWLOW model by concentrating on balanced truncation technique which is appropriate for medium size systems. In addition, the proposed method either guarantees or provides guaranteed stability in some mentioned conditions. Finally, applicability of the reduced model, instead of high-order decoupled multi-model in weighting multi-model controllers, is investigated through the control of a nonlinear Alstom gasifier benchmark.     

Frequency Interval Gramians Based Structure Preserving Model Order Reduction for Second Order Systems

A new structure preserving model order reduction technique for second order systems in limited frequency interval is presented. Frequency limited Gramians (FLGs) and corresponding continuous time algebraic lyapunov equations (CALEs) are developed. For solution of CALEs and Cholesky factorization of FLGs, computationally efficient approximation scheme is proposed. Multiple transformations based on balancing of frequency limited position or velocity Gramians are defined in order to compute Hankel singular values (HSVs). Frequency limited second order balanced truncation based on magnitudes of HSVs is performed for order reduction. Moreover, stability conditions for reduced order models (ROMs) are stated and algorithms for achieving stability in ROMs are proposed. Results are compared with existing technique to certify the usefulness of the proposed technique.     

WANG氏RC互连线模型平衡截断化简的分解算法

平衡截断（Balanced Truncation）是一种有效的模型简化（Model Reduction）方法,它的优点是简化模型有一个误差上限,使简化模型的性能得到保证.该文介绍一种平衡实现的分解算法,并应用Matlab对一4阶的Wang氏RC互连线模型的平衡实现及其模型截断简化进行编程仿真.单位阶跃响应和波特图显示原始模型与平衡模型曲线重合.给出了1阶、2阶、3阶平衡截断简化模型的阶跃响应和波特图,并对简化模型的理论误差上限与模型的实际最大误差进行了对比.方法可用于对IC互连线等模型的简化.