Model reduction by phase matching

This paper discusses procedures for approximating high-order rational power spectrum matrices and minimum phase stable transfer function matrices by lower-order objects of the same type. The basis of the approximation is to secure closeness of a high-order and low-order minimum phase stable transfer function matrix in phase, and to infer from this, closeness in magnitude. A suitable definition of multivariable phase is needed. Particular cases of the approximation procedure which are already known are cast in a general framework, which is also shown to include relative error approximation. A number of error bounds are given. Extensions to approximation of nonminimum phase transfer function matrices are also provided.     

Two-step procedure for time-suboptimal control

A two-step procedure for stable time-suboptimal control is proposed. The procedure involves the use of two feedback gain matrices. The first matrix is determined to transfer the system from a given state to the vicinity of the desired state in minimum time. Then a switch is made to a second feedback gain matrix obtained from shifting of eigenvalues For maximum stability. The viability of the proposed procedure is demonstrated with a linear six-p!ate gas absorber and a non-linear two-stage CSTR system.     

Routh approximations for reducing order of linear, time-invariant systems

A new method of approximating the transfer function of a high-order linear system by one of lower order is proposed. Called the "Routh approximation method" because it is based on an expansion that uses the Routh table of the original transfer function, the method has a number of useful properties: if the original transfer function is stable, then all approximants are stable; the sequence of approximants converge monotonically to the original in terms of "impulse response" energy; the approximants are partial Padé approximants in the sense that the firstkcoefficients of the power series expansions of thekth-order approximant and of the original are equal; the poles and zeros of the approximants move toward the poles and zeros of the original as the order of the approximation is increased. A numerical example is given for the calculation of the Routh approximants of a fourth-order transfer function and for illustration of some of the properties.     

Approximation of multivariable linear systems with impulse response and autocorrelation sequences

This paper considers the construction of approximants of multi-input-multi-output, discrete-time linear systems from the finite data of the impulse response and autocorrelation sequences. In the approximation of a multivariable linear system, it is common practice to use a finite portion of its impulse response sequence. This is formally equivalent to the Padé approximation technique, which may produce unstable approximants, even though the original system is stable. Mullis and Roberts proposed a new method, which yields stable approximants, in connection with approximation of digital filters. This is, however, restricted to the single-input-single-output case. This paper extends their method to the multi-input-multi-output case and shows a fast recursive algorithm to construct stable approximants of linear systems.     

一种新型的非参数模型及其在系统辨识中的应用

本文给出了一个求解广义正交多项式的微分运算矩阵的新方法，应用对连续线性系统的脉冲响应函数进行正交逼近的方法来讨论脉冲响应函数的实现问题，得到了一类新型的非参数模型，并导出了利用该模型来辨识连续线性系统的脉冲响应函数的算法，最后给出了例子证实本文所给方法的有效性。     

H∞-norm approximation of systems by constantmatrices and related results

In this work the H_∞-norm approximation of a given stable proper rational transfer function by a constant matrix is considered. A new improved bound on H_∞-norm of a given stable transfer function is derived. Optimal approximation for relaxation systems is given     

On linear system reduction by impulse energy approximation

Linear system reduction by impulse response energy approximation is analyzed and limitations are discussed. To show that in general the claims made for the procedure are not true, a numerical example is included.     

Legendre orthogonal polynomials approximation of system gramians and its application to balanced truncation

The paper presents a procedure for computation of system gramians by using Legendre orthogonal polynomials approximation of the system state impulse and output responses. The proposed approach is trajectory based and relies on the system state and output trajectories snapshots selected either by experiment or computer simulation. It is defined in deterministic settings as opposed to similar approaches defined in stochastic settings by using the data covariance matrix. The advantage of using orthogonal series approximation for the gramians is to avoid solving the usual Lyapunov equations. The proposed method can be equally well applied to linear time-invariant as well as time-varying systems, and even to unstable systems, since the gramians approximation is performed on a finite interval of time. When the observation interval contains the whole energy of the system state impulse and output responses, the proposed method gives similar results as the gramians computed by solving Lyapunov equations. Several experiments are performed showing the good approximation properties of the presented method.     

Initial-value solutions for on-line smoothing and filtering of linear stationary signals

This note presents an initial-value solution for the impulse response function specifying the linear least-squares estimates of a linear scalar stationary signal with general covariance functions in the presence of a white Gaussian noise. The method employed is an invariant imbedding. The resulting initial-value solution enables us to design a new on-line filter, and fixed-point, initial-point, and fixed-lag smoothers based on covariance specification.     

An extended inner–outer factorisation algorithm based on the structure of a transfer function matrix inverse

In this paper, the structure feature of the inverse of a multi-input/multi-output square transfer function matrix is explored. Instead of complicated advanced mathematical tools, we only use basic results of complex analysis in the analysing procedure. By employing the Laurent expression, an elegant structure form of the expansion is obtained for the transfer function matrix inverse. This expansion form is the key of deriving an analytical solution to the inner–outer factorisation for both stable plants and unstable plants. Different from other computation algorithm, the obtained inner–outer factorisation is given in an analytical form. The solution is exact and without approximation. Numerical examples are provided to verify the correctness of the obtained results.     

基于脉冲响应的输出误差模型的辨识

基于系统脉冲响应参数, 利用相关分析方法, 提出了一种辨识输出误差模型参数的方法. 该方法是利用有限脉冲响应模型逼近输出误差模型, 通过依次递增脉冲响应参数的数目N来提高逼近精度. 理论分析表明, 只要N足够大, 模型的辨识精度可以满足实际要求. 提出的辨识方法可以在假设阶次N =1的条件下, 依次递增计算N较大时的脉冲响应参数和目标函数值, 从而根据脉冲响应确定系统的参数. 仿真试验说明提出的方法估计输出误差模型的参数是有效的.     

Structured low-rank approximation and its applications

Fitting data by a bounded complexity linear model is equivalent to low-rank approximation of a matrix constructed from the data. The data matrix being Hankel structured is equivalent to the existence of a linear time-invariant system that fits the data and the rank constraint is related to a bound on the model complexity. In the special case of fitting by a static model, the data matrix and its low-rank approximation are unstructured.We outline applications in system theory (approximate realization, model reduction, output error, and errors-in-variables identification), signal processing (harmonic retrieval, sum-of-damped exponentials, and finite impulse response modeling), and computer algebra (approximate common divisor). Algorithms based on heuristics and local optimization methods are presented. Generalizations of the low-rank approximation problem result from different approximation criteria (e.g., weighted norm) and constraints on the data matrix (e.g., nonnegativity). Related problems are rank minimization and structured pseudospectra.     

A Constructive Approach to Approximate Linear Periodic Systems

In this note, the problem on approximating a linear periodic system with period N by a periodic model with period M(0infin norms. Under this framework, both finite impulse response (FIR) and infinite impulse response (IIR) periodic systems can be efficiently approximated by using a constructive linear matrix inequality approach. Finally, some numerical examples are given to illustrate the effectiveness of the proposed approach     

Recursive inversion of externally defined linear systems by FIRfilters

The approximate inversion of an internally unknown linear system, given by its impulse response sequence, by an inverse system having a finite impulse response, is considered. The recursive least-squares procedure is shown to have an exact initialization based on the triangular Toeplitz structure of the matrix involved. The proposed approach also suggests solutions to the problem of system identification and compensation     

A connection between normalized coprime factorizations and linear quadratic regulator theory

Given a transfer matrix described by a minimal state-space triple, a method is given for computing state-space realizations for the numerator and denominator of a normalized, stable, right coprime factorization for the transfer matrix. The method involves the solution of an algebraic Riccati equation. It allows the use of existing computational state-space algorithms in finding normalized stable right coprime factorizations, and avoids explicit calculations of spectral factors.     

Model following using partial state feedback

This short paper considers the problem of designing a compensating control scheme for an observable linear multivariable plant using partial state feedback so that the impulse response matrix of the resulting system exactly corresponds to the impulse response matrix of a prespecified linear model. A procedure is given for the synthesis of the compensator that does not require a separate realization for the plant observer. A sufficient condition is given for the internal stability of the model following system.     

Rational approximation of a class of infinite-dimensional systems II: Optimal convergence rates ofL ∞ approximants

The achievable errors in infinity-norm approximation of an irrational transfer function by a rational one of given degree are considered. Error bounds are given which have particular application to delay systems, and it is shown that optimal convergence rates are achievable if the corresponding impulse response has certain smoothness properties.     

一种新型FPGA器件延时计算方法

在深亚微米工艺条件下,被广泛使用的Elmore模型明显高估FPGA互连线延时;通过对RC电路冲激响应的研究,提出了采用前3阶矩确立主极点模型来计算FPGA连线延时的方法;该方法实现了计算精度和计算复杂性的折中,理论上证明该方法适用于任何结构RC电路,并且小于Elmore延时;实验表明,该方法对于远端节点估计的延时值和Spice仿真值相差不到1％;应用于商用FPGA,计算所得互连线延时的平均误差小于Elmore模型的三分之一。     

Prediction error identification of linear systems: A nonparametric Gaussian regression approach

A novel Bayesian paradigm for the identification of output error models has recently been proposed in which, in place of postulating finite-dimensional models of the system transfer function, the system impulse response is searched for within an infinite-dimensional space. In this paper, such a nonparametric approach is applied to the design of optimal predictors and discrete-time models based on prediction error minimization by interpreting the predictor impulse responses as realizations of Gaussian processes. The proposed scheme describes the predictor impulse responses as the convolution of an infinite-dimensional response with a low-dimensional parametric response that captures possible high-frequency dynamics. Overparameterization is avoided because the model involves only a few hyperparameters that are tuned via marginal likelihood maximization. Numerical experiments, with data generated by ARMAX and infinite-dimensional models, show the definite advantages of the new approach over standard parametric prediction error techniques and subspace methods both in terms of predictive capability on new data and accuracy in reconstruction of system impulse responses.