Author's reply [to comments on 'On Routh-Pade model reduction of interval systems']

The paper is reply to the article by Shih-Feng Yang (see ibid., p.273-4, 2005). Previously (Dolgin and Zeheb, 2003), it was shown that the existing generalization of the direct Routh table truncation method for interval systems fails to produce a stable system. Additionally, it was shown how to extend the method to ensure the stability of the resulting interval system. Although the main statement of Dolgin and Zeheb (2003) is correct, there is, however, an omission in the proposed algorithm of building interval Routh table:the newly calculated line in the table may be inconsistent with the last existing line. Two additional conditions should be formulated to avoid this situation. The main idea is to shrink the uncertainty of the elements of the last existing line of the table, in the procedure of building the new line.     

Routh-Pade approximation for interval systems

Presents a method for the reduction of the order of interval system. The denominator of the reduced model is obtained by a direct truncation of the Routh table of the interval system. The numerator is obtained by matching the coefficients of power series expansions of the interval system and its reduced model. A numerical example illustrates the proposed procedure     

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     

Stable biased reduced order models using the Routh method of reduction

A generalization of the Routh method of reduction is introduced for obtaining stable reduced order models. The reduced models may be ‘ biased ’ in the sense that they may approximate the initial transient response of the high order system more closely than the steady-state response, and vice-versa. Given the desired order of the reduced model, the method of this paper produces a number of stable reduced models which approximate the high order system. The method is easily extended to multi-variable systems. Examples are given to illustrate the method and to make comparisons with other methods of reduction.     

Continued fraction expansion and inversion by ‘rearranged’ Routh tables

The Routh algorithm is known to be the simplest method for continued fraction expansion and inversion but it faces a serious limitation when the first column entry of any row of the Routh table becomes zero (but not a zero row). In this paper a remedy for such situations is proposed by rearranging the coefficients of the row in which zero entry has occurred in the first column of the Routh table. All possible cases, where the first column entry of the Routh table may become zero, are discussed for Cauer's first and second forms and are illustrated by numerical examples. The coefficients of the rearranged Routh table are also calculated by the usual Routh algorithm, hence it is equally suitable for digital computation.     

Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems

A Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems is presented via an auxiliary integer degree polynomial. The presented Routh test is a classical Routh table test on the auxiliary integer degree polynomial derived from and for the commensurate fractional degree polynomial. The theoretical proof of this proposed approach is provided by utilizing Argument principle and Cauchy index. Illustrative examples show efficiency of the presented approach for stability test of commensurate fractional degree polynomials and commensurate fractional order systems. So far, only one Routh-type test approach [1] is available for the commensurate fractional degree polynomials in the literature. Thus, this classical Routh-type test approach and the one in [1] both can be applied to stability analysis and design for the fractional order systems, while the one presented in this paper is easy for peoples, who are familiar with the classical Routh table test, to use.     

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     

What can routh table offer in addition to stability？

Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a litde beyond , the number of unstable roots of a polynomial in terms of the signs of certain entries of the Routh table constructed from the coefficients of the polynomial. The use of the Routh table, as far as the common textbooks show, is only limited to this function. We will show that the Routh table can actually be used to construct an orthonormal basis in the space of strictly proper rational functions with a common stable denominator. This orthonormal basis can then be used for many other purposes, including the computation of the H2 norm, the Hankel singular values and singular vectors, model reduction, H∞ optimization, etc.     

Control Routh Array And Its Applications

In this paper the Routh stability criterion [16] has been developed into control Routh array. Some formulas for calculating the array are provided. Certain properties are investigated. Using it, the problem of stabilization of bilinear systems via constant controls is solved. Then by converting the problems, the method of control Routh array is also used to solve some other stability related problems, such as the stabilization of control systems, the stability of uncertain Hurwitz matrix and the stability of interval matrices. Several algorithms have been developed to provide numerical solutions.     

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.     

A novel approach for root distribution analysis of linear time‐invariant systems using Routh and Fuller tables

The root distribution of a given characteristic equation of a linear time‐invariant system can be analyzed with the help of a Routh table using the elements of the first column in the table. In the case of unstable systems, sometimes, a zero element may appear in the third row of the first column of the Routh array. This prematurity can be suitably handled as indicated by various authors. In this paper, the given characteristic polynomial having roots in the right hand plane is multiplied by a suitable polynomial, and Routh and Fuller tables are applied for the resultant polynomial to infer the complete root distribution. Further, the column polynomials from each table are adopted to know more about root distribution, which forms the core of the proposed work. The Routh table helps in counting and locating roots in the s‐plane, and the Fuller table helps in depicting whether the roots are distinct or complex in nature. In this regard, it is shown in this paper that the simultaneous integration of Routh and Fuller tables yields a good amount of information regarding the root distribution in the s‐plane. The newly presented procedure is illustrated with examples. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Model Reduction of Nonstationary LPV Systems

This paper focuses on the model reduction of nonstationary linear parameter-varying (NSLPV) systems. We provide a generalization of the balanced truncation procedure for the model reduction of stable NSLPV systems, along with a priori error bounds. Then, for illustration purposes, this method is applied to reduce the model of a two-mass translational system. Furthermore, we give an approach for the model reduction of stabilizable and detectable systems, which requires the development and use of coprime factorizations for NSLPV models. For the general class of eventually periodic LPV systems, which includes periodic and finite horizon systems as special cases, our results can be explicitly computed using semidefinite programming     

Frequency-domain reduction of linear systems using Schwarz approximation

A frequency domain approach for reducing linear, time-invariant systems is presented which produces stable approximations of stable systems. The method is based upon the Schwarz canonical form and is shown to have a continued-fraction representation. A link between this method and the popular Routh approximation is also given. Further, the Schwarz approximation is combined with a moments-matching technique to improve steady-state responses to step and polynomial inputs. Examples are given to illustrate the methods.     

A simple and direct method of reducing order of linear systems using routh approximations in the frequency domain

A simple and direct method of approximating higher order systems by lower order ones based on Routh approximants without using reciprocal transformations is proposed. New algorithms for constructing reduced order transfer functions are presented. A method of approximating unstable systems with poles in RHP is also proposed resulting in unique transfer functions.     

平面系统稳定性的间接方法*

利用Routh 判据得到的间接检验矩阵特征值的方法讨论平面系统的稳定性问题,对于线性系统,该方法给出平面区间矩阵稳定性的充要条件.对于非线性系统,Jacobian 猜想的已知结果[5],该方法给出平面非线性系统稳定吸引域估计的一种算法.最后,该估计被利用于高维系统二维滑动模的设计.     

平面系统稳定性的间接方法

利用Ｒｏｕｔｈ判据得到的间接检验矩阵特征值的方法平面系统的稳定性问题。对于线性系统,该系统法给出平面区间矩阵稳定性的充要条件。对于非线性系统,基于Ｊａｃｂｉａｎ猜想的已知结果。     

Balanced truncation for unstable infinite dimensional systems via reciprocal transformation

We discuss a model reduction for unstable infinite dimensional systems using balanced truncation of reciprocal systems. The systems considered are assumed to be exponentially stabilizable and detectable, with bounded and finite rank input and output operators. We decompose the systems into their stable and unstable parts and perform reciprocal transformation only on the stable part. Balanced truncation is carried out on the reciprocal systems. This yield is then translated using reciprocal transformation as the reduced-order model of the stable part. Finally, we added the unstable part to the reduction of the stable part as the overall reduced-order model for the systems. To verify the effectiveness of the proposed approach, numerical simulations are applied to the heat equation. The performance of the proposed reduction method is compared to the balanced truncation method.     

γ-δ Routh approximation for interval systems

The paper presents the γ-δ Routh approximation for interval systems. The interval γs and δs are evaluated for the higher order interval systems, and then an rth-order approximant is obtained by retaining the first r, interval δs, and γs. A numerical example illustrates the procedure     

Implications of Routh stability criteria

According to the Routh criteria, a polynomial is stable if and only if all the coefficients in the first column of the stability table are of the same sign. Does this imply that elements in other columns would be of the same sign? This is investigated here in detail. The investigation reveals that if any element in any row of the table is of different sign, the system is unstable.     

Singular perturbation approximation of balanced systems

This paper relates the singular perturbation approximation technique for model reduction to the direct truncation technique if the system model to be reduced is stable, minimal and internally balanced. It shows that these two methods constitute two fully compatible model-reduction techniques for a continuous-time system, and both methods yield a stable, minimal and internally balanced reduced-order system with the same L_∞-norm error bound on the reduction. Although the upper bound for both reductions is the same, the direct truncation method tends to have smaller errors at high frequencies and larger errors at low frequencies, while the singular perturbation approximation method will display the opposite character. It also shows that a certain bilinear mapping not only preserves the balanced structure between a continuous-time system and an associated discrete-time system, but also preserves the slow singular perturbation approximation structure. Hence the continuous-time results on the singular perturbation approximation of balanced systems are easily extended to the discrete-time case. Examples are used to show the compatibility and the differences in the two reduction techniques for a balanced system