Routh Approximation Research Articles

Reduction of Linear Time-Invariant Systems Using Routh-Approximation and PSO

Order reduction of linear-time invariant systems employing two methods; one using the advantages of Routh approximation and other by an evolutionary technique is presented in this paper. In Routh approximation method the denominator of the reduced order model is obtained using Routh approximation while the numerator of the reduced order model is determined using the indirect approach of retaining the time moments and/or Markov parameters of original system. By this method the reduced order model guarantees stability if the original high order model is stable. In the second method Particle Swarm Optimization (PSO) is employed to reduce the higher order model. PSO method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. Both the methods are illustrated through numerical examples.

Reduction of Power System Model Using Balanced Realization, Routh and Pade Approximation Methods

A time domain model reduction technique based on balanced realization has been applied to a single machine power system model i.e., a synchronous machine connected to an infinite bus through transmission line. The controllability and observability grammians have been computed by solving the Lyapunov equations. The high order system is transformed into balanced form by using the similarity transformation matrix to bring the grammian matrices to the balanced format, which is used to formulate the reduced order model in state space form. The results have also been compared with the Routh and Pade approximation methods, the two well-known frequency domain techniques. The comparison of time and frequency responses shows that, the reduced order models satisfactorily approximate the original system.

IMPROVEDγ – δ ROUTH APPROXIMATION OF INTERVAL SYSTEMS USING INTERVAL ARITHMETIC

In γ – δ Routh approximation of interval systems, interval Routh approximant may not be robustly stable even if the original interval system is robustly stable. Thus, in this paper an improvement is proposed to the γ – δ Routh approximation for interval systems using Kharitonov polynomials such that the resulting interval Routh approximant is always robustly stable. A numerical example illustrates the procedure.

Suboptimal Bilinear Routh Approximant for Discrete Systems

Recently an improved bilinear Routh approximation method has been suggested for the order reduction of discrete systems. In the method, the last α and β parameters of a reduced model were replaced by new parameters so that the impulse response energy of an original system is also preserved in the reduced model without destroying the stability preserving and time-moments matching properties. In this paper a new and simple improvement is proposed from which one can find a suboptimal bilinear Routh approximant. Compared to the previous result, the approach of this paper has an advantage that the improvement is always guaranteed.

Optimal Routh approximants through integral squared error minimisation: computer-aided approach

A computer-aided method for obtaining a reduced-order approximant of a given (stable) single-input single-output system based on the minimisation of integral squared error (ISE) pertaining to a unit-step input is presented. Both the numerator and denominator coefficients of the model are treated as free parameters in the process of optimisation. The method has a built-in stability-preserving feature. The problem of formulating the ISE is circumvented by introducing a set of equality constraints.

A New Method for Modelling of Large Scale Interval Systems

In recent years, simulation, design and control of systems with parametric uncertainties for the entire range of operation gained much importance. This paper presents a new method for reducing order of Large Scale Interval systems. The proposed method involves simple computations, and is efficient as it needs the formulation of only one Routh type tabulation, avoiding the necessity of formulating two tables viz., γ and δ tables, and avoids the computation of time moments of the original high order Interval system beforehand, unlike other available methods [1,2]. The proposed Simplified Routh Approximation method (SRAM) always generates stable reduced order Interval systems, retaining the initial time moments of the original high order system. The proposed simplified method is extended for the reduction of high order Multivariable and Discrete Interval systems to overcome the limitations and drawbacks of some of the existing methods [1,2] in literature. Typical numerical examples are considered to illustrate the flexibility and effectiveness of the proposed method.

Equivalence of bilinear Routh and Schwarz approximation methods for discrete-time systems

Two methods using the bilinear Routh approximations and Schwarz approximations have been proposed in the literature for the order reduction of discrete-time systems. Both methods yield the reduced models which preserve the stability and first few time-moments of the original ones. Recently, improvements have been made so that the impulse response energies of original models are also retained in the reduced models. In this Letter, it is shown that the former two methods and their improved versions are respectively equivalent to each other.

Optimal model reduction of discrete-time descriptor systems

An optimal model reduction method is presented to obtain stable reduced-order models for discrete-time descriptor systems. A parametrization based on the bilinear Routh approximation of linear normal discrete-time systems is used to parametrize the causal subsystems of the reduced-order models. The expressions for the error and its gradient are explicitly given. They are then employed to solve an unconstrained optimization problem for the model reduction problem. The descriptor system structure is preserved in the reduced-order models.

Optimal model reduction of discrete-time descriptor systems

An optimal model reduction method is presented to obtain stable reduced-order models for discrete-time descriptor systems. A parametrization based on the bilinear Routh approximation of linear normal discrete-time systems is used to parametrize the causal subsystems of the reduced-order models. The expressions for the error and its gradient are explicitly given. They are then employed to solve an unconstrained optimization problem for the model reduction problem. The descriptor system structure is preserved in the reduced-order models.

Pade type model order reduction for multivariable systems using routh approximation

A method of obtaining reduced order models for multivariable systems is described. It is shown that the method has several advantages, e.g. the reduced order models retain the steady-state value and stability of the original system. Irrespective of whether the original multivariable system is described in state space form or in the transfer matrix form, the proposed method yields the reduced order models in state space form. In this method the Routh approximation is used to formulate the common denominator polynomial of a reduced order model. This is used to describe the structure of A r matrix. The matrices B r and C r are chosen appropriately and some of the elements of B r / C r matrices are specified in such a way that after matching time moments/Markov parameters, the resulting equations are linear in the unknown elements of B r and C r matrices. The procedure is illustrated via a numerical example.

Large scale interval system modellingusing Routh approximants

The authors present a new method for reducing the order of large scale interval systems using Routh approximants. The method is computationally simple, requires only the γ table and avoids the need to formulate both γ - δ tables and evaluate the time moments of the original system in advance unlike in other methods. The method always generates low-order models, which retain the stability and initial time moments of the original high-order interval system. Typical numerical examples illustrating the proposed method are provided.

Improved Bilinear Routh Approximation Method for Discrete-Time Systems

Hwang and Shieh proposed a bilinear Routh approximation method for reducing the order of discrete-time systems. A reduced model derived by the method is not only stable whenever an original model is stable, but also fits the first few time-moments of the original one. This paper addresses the possibility of improving the method by letting the impulse response energy of the original model also be conserved in the reduced model without destroying the stability preserving and time-moments matching properties.

Direct method for obtaining modified Routh approximants

To obtain a kth-order reduced order model by the modified Routh approximation method, it is necessary to compute all reduced models with degree less than k. A direct algorithm for obtaining a kth-order reduced model without calculating those lower-order models is presented.

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.

Improvement to modified Routh approximation method

The modified Routh approximation method yields a stable reduced model which maintains the first few time-moments and Markov parameters of an original model. An improvement is made so that the impulse energy of the original model is also preserved in the reduced model.

A new family of Routh approximants

Based on the combinatorial Routhα-β andγ-δ expansions of a stable transfer function, a new energy decomposition tree for linear systems is developed. The pertinent properties to the energy decomposition tree are investigated, and an algorithm is derived for synthesizing transfer functions from the tree. The synthesis process naturally leads to a new family of Routh approximants to the system. It is indicated that the selection of Routh approximants based on the values of impulse-response energy is often inadequate because there may be a number of different Routh approximants with the same order and the same impulse-response energy. In such cases, an additional performance criterion, such as the integral of squared error of impulse response or unit-step response, has to be used to select a suitable Routh approximant.

γ-δ Routh approximation for interval systems

The paper presents the /spl gamma/-/spl delta/ Routh approximation for interval systems. The interval /spl gamma/s and /spl delta/s are evaluated for the higher order interval systems, and then an rth-order approximant is obtained by retaining the first r, interval /spl delta/s, and /spl gamma/s. A numerical example illustrates the procedure.

A proof of stability of all‐pass PADé approximants TOe‐s

Abstract This paper presents an explicit derivation of the all‐pass Routh approximants to e‐s. It then shows that the stable all‐pass Routh approximants to e‐s are exactly the same as the all‐pass Padé approximants. As a result, the derivation provides an alternative way of proving that the all‐pass Padé approximants to e‐s are always asymptotically stable.

Multifrequency Routh approximants for linear systems

A multipoint Routh gamma - delta canonical continued-fraction expansion for the transfer function of a linear system is derived. Based on this general form, a multifrequency Routh approximant to the system is derived by selecting the expansion points on the imaginary axis and truncating the resulting continued-fraction expansion. A connection between the stability preservation property of the multifrequency Routh approximants and the expansion points on the imaginary axis is established. Thus the multifrequency Routh approximation procedure is flexible in deriving stable reduced-order models while fitting the frequency responses and retaining the time moments and/or Markov parameters of the impulse response of the system.

An Improved Approach for Biased Model Reduction using Impulse Energy Approximation Technique

A new method for biased model reduction of high order linear, time-invariant systems is proposed. This paper presents a new method for generating reduced order models that retain some initial time moments and Markov parameters of the high order systems (biased models). The proposed method is based on approximating the impulse energy value of the reduced order model with that of the original system. This advantageous feature guarantees a better impulse response approximation over the existing Routh approximation methods. New algorithms are proposed to generate low order models to retain the stability using only α parameters unlike other methods which need both the α and β parameters. A better contribution to the impulse response energy of the κth order reduced model is made by an appropriate selection of these parameters. The effectiveness and the flexibility of the proposed method is illustrated by two typical numerical examples.