A Parametrized Controller Reduction Technique via a New Frequency Weighted Model Reduction Formulation

In this technical note, the solution to the controller reduction problem via a double-sided frequency weighted model reduction technique is considered. A new method for finding low-order controllers based on new frequency weights derived using closed-loop system approximation criterion is proposed. The formulas of frequency weights are obtained in terms of the plant, the original controller and a matrix of free parameters. By varying the free parameters in the resulting two-sided frequency weighted model reduction problem, frequency weighted error can be significantly reduced to yield more accurate low-order controllers.      

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).     

Frequency-weighted model reduction via quasi-Kalman decomposition

A frequency-weighted model reduction method based on quasi-Kalman decomposition is proposed for discrete systems. The method is essentially an extension of the technique presented by Lu and Lee (1985) to the frequency weighted case and can handle both single-sided as well as double-sided weights. Two illustrative examples show the effectiveness of the method and results are compared with the method developed by Enns (1984)     

Factorization‐based frequency‐weighted optimal Hankel‐norm model reduction

In this paper, we present frequency‐weighted optimal Hankel‐norm model reduction algorithms for linear time‐invariant continuous‐time systems by representing an original higher‐order system into new fictitious systems. The new system representations are derived through factorization of the resulting sub‐matrices that are obtained after transformations. As the proposed approaches are factorization dependent, additional results with both approaches are included using another factorization of the fictitious input–output and weight matrices. The proposed algorithms generate stable reduced models with double‐sided weights and provide a substantial improvement in the weighted error. A numerical example is given to compare the efficacy of the proposed algorithms with the well‐known frequency‐weighted techniques.     

Model reduction via balanced realizations: an extension andfrequency weighting techniques

Two model-reduction methods for discrete systems related to balanced realizations are described. The first is a technique which utilizes the least controllable and observable subsystem in deriving a balanced discrete reduced-order model. For this technique as L ^∞ norm bound on the reduction error is given. The second method is a frequency-weighting technique for discrete- and continuous-time systems where the input-normal or output-normal realizations are modified to include a simple frequency weighting. For this technique, L^∞ norm bounds on the weighted reduction errors are obtained     

Identification and model reduction from impulse response data

Two new algorithms for identification and model reduction of stable linear continuous systems are proposed, based on the weighted impulse response gramians (Agathoklis and Sreeram 1988 b). In identification, the model parameters are obtained from the solution of a linear system of equations with coefficients obtained from the numerical evaluation of the weighted impulse response gramians. The reduction technique is based on retaining part of the original weighted impulse response gramians obtained as the solutions to the Lyapunov equation for the original system in controllability canonical form. This yields different stable models for different values of the weighting factor. The model corresponding to zero weighting factor matches the impulse response norm of the original system and its derivatives exactly. Finally, the method is illustrated by a numerical example and is compared with well-known balanced realization techniques.     

Computationally Efficient Algorithm For Frequency‐Weighted Optimal H∞ Model Reduction

In this paper, a frequency‐weighted optimal H_∞ model reduction problem for linear time‐invariant (LTI) systems is considered. The objective of this class of model reduction problems is to minimize H_∞ norm of the frequency‐weighted truncation error between a given LTI system and its lower order approximation. A necessary and sufficient solvability condition is derived in terms of LMIs with one extra coupling rank constraint, which generally leads to a non‐convex feasibility problem. Moreover, it has been shown that the reduced‐order model is stable when both stable input and output weights are included, and its state‐space data are given explicitly by the solution of the feasibility problem. An efficient model reduction scheme based on cone complementarity algorithm (CCA) is proposed to solve the non‐convex conditions involving rank constraint.     

Model reduction using impulse-response gramians: a frequency-domain approach

A frequency domain model reduction technique based on the impulse-response gramian is proposed. Two new methods for evaluation of the impulse-response gramian in the frequency domain are also presented. The Routh technique relies on a Routh table to evaluate energy integrals of the type found on the impulse-response gramian diagonal, while the second approach uses an Inners determinant technique. Off diagonal elements are computed via system Markov parameters and knowledge of diagonal values. The model reduction technique, involving truncation of the impulse-response gramian, is a variation on that presented by Agathoklis and Sreeram (1990 a). The proposed method evaluates the transfer function of the reduced-order model directly rather than producing a state space representation. Algorithms outlining the steps involved in impulse-response gramian evaluation, plus those for model reduction, are given. Each is supported by a numerical example     

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.     

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.     

Optimal Model Reduction with a Frequency Weighted Extension

Inthis paper, a model reduction technique based on optimizationis presented. The objective function minimized is the impulseenergy of the overall system. An extension of the technique tothe frequency weighted case is also presented, where single-sidedor double-sided weightings can be incorporated in the reductionprocedure. The paper proposes an alternative to find an optimizationsolution by solving ordinary differential equations which aregradient flow associated with the objective function to be minimized.Two examples are presented to illustrate the effectiveness ofthe method.     

A Survey of Model Reduction by Balanced Truncation and Some New Results

Balanced truncation is one of the most common model reduction schemes. In this note, we present a survey of balancing related model reduction methods and their corresponding error norms, and also introduce some new results. Five balancing methods are studied: (1) Lyapunov balancing, (2) stochastic balancing, (3) bounded real balancing, (4) positive real balancing and (5) frequency weighted balancing. For positive real balancing, we introduce a multiplicative-type error bound. Moreover, for a certain subclass of positive real systems, a modified positive-real balancing scheme with an absolute error bound is proposed. We also develop a new frequency-weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians. Two numerical examples are illustrated to verify the efficiency of the proposed methods.     

Principle of frequency-domain balanced structure in linear systems and model reduction

In this paper a novel model reduction technique for linear time-invariant systems is presented. The proposed technique is based on a conceptual viewpoint regarding the balancing of the controllability and observability Gramians of a multivariable system in a given range of frequency. The conditions for the stability of the reduced model are also provided. From a real-time applicability viewpoint, the frequency-domain balanced structure provides an attractive approach to the model reduction of large scaled systems. The simulation results establish the effectiveness of this proposed method compared to the effectiveness of existing techniques.     

Frequency-domain generelaized singular peruturbation method for relative error model order reduction

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in ttiis paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.     

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.     

Performance preserving controller reduction via additiveperturbation of the closed-loop transfer function

An H_∞ performance preserving controller order reduction method is proposed. Here performance preservation indicates that the H_∞ norm bound of the closed loop transfer function with reduced-order controller is not greater than the H_∞ norm bound of the closed loop transfer function with full order controller. We assume additive perturbations to the closed-loop transfer function and obtain a sufficient condition for performance preservation. Two kinds of useful weightings are derived, and the controller reduction problem is solved via a frequency weighted model reduction problem     

基于Hankel范数模型降价的控制对象的名义模型选择

研究了控制对象具有多个模型时,求取其适合鲁棒控制器设计的名义模型的问题. 提出了一种基于Hankel范数模型降阶的名义模型选择算法.仿真结果确认了算法的有效 性.     

一种基于加权TOPS的宽带DOA估计新方法

针对投影子空间正交性测试（Test of orthogonality of projected subspaces,TOPS）对宽带信号波达方向估计（Direction?of?arrival, DOA）存在角度分辨率较低,且易出现伪峰的问题,提出了一种加权TOPS的宽带DOA估计新方法。该方法通过最大化各频率点信号子空间与噪声子空间特征值区分度选择参考频点,同时利用信号子空间投影代替其零空间投影;然后利用正交频率子空间测试法（Test of orthogonality of frequency subspaces, TOFS）对平方TOPS法的判定矩阵进行加权修正;最后对判定矩阵求迹实现宽带DOA估计,避免了奇异值分解。与现有的TOPS法、平方TOPS以及TOFS相比,该方法提高DOA估计精度,能够有效剔除伪峰,降低了算法复杂度,且对间隔相近信源DOA估计分辨率更高。仿真实验结果验证了该方法的有效性。     

Improved results on frequency‐weighted balanced truncation and error bounds

In this paper, we present some new results on frequency‐weighted balanced truncation which is a significant improvement on Lin and Chiu's frequency‐weighted balanced truncation technique. The reduced‐order models, which are guaranteed to be stable in the case of double‐sided weighting, are obtained by direct truncation. Two sets of simple, elegant and easily calculatable a priori error bounds are also derived. Numerical examples and comparison with other well‐known techniques show the effectiveness of the proposed technique. Copyright © 2011 John Wiley & Sons, Ltd.