Routh Table Research Articles

ПРОЕКТИРОВАНИЕ СИСТЕМЫ УПРАВЛЕНИЯ ГУСЕНИЧНЫМ РОБОТОМ С ПОМОЩЬЮ ПАКЕТА MATLAB

When designing a feedback control system, the key issue is to ensure its stability. The stability of a feedback system is directly related to the location of the roots of its characteristic equation. The rules that allow to determine whether the system is stable or not without calculating the roots of the characteristic equation are important. In this paper, a very useful method of stability analysis is considered, known as the Routh method. This method is most simply explained by the Table he proposed. The tracked robot control system is stable when all elements of the first column of the Routh Table are positive. If there is at least one negative element in the first column, then the system is unstable. The stable region for the microprocessor system of a tracked robot was also found using the Simulink model. Within the framework of this study, the movement of the robot in a circle (1st trajectory) and uniform rectilinear movement (2nd trajectory) were considered. The steady-state error is calculated for both the 1st trajectory and the 2nd one. It is shown that the steady-state error for the 1st trajectory is equal to zero, i.e. the tracked robot accurately tracks the given input signal. It is shown that the steady-state error for the 2nd trajectory does not exceed 24% of the value of the linear input signal. A microprocessor system has been developed to control the movement of a tracked robot, where the adjustable parameters are K and a. To obtain a numerical result in modeling the stability of a tracked robot, the MATLAB package was used.

Model Order Diminution of Discrete Interval Systems Using Kharitonov Polynomials

In this research proposal, diminution of higher order (HO) discrete interval system (DIS) is accomplished by utilizing Kharitonov polynomials. The DIS is firstly, transformed into continuous interval system (CIS). The Markov-parameters (MPs) and time-moments (TMs) are exploited for determination of approximated models. The ascertainment of model order diminution (MOD) of DISs is done by Routh-Padé approximation. The Routh table is utilized to obtain the denominator of approximated model. The unknown numerator coefficients of desired approximated model are determined by matching MPs and TMs of DISs and desired model. This whole procedure of MOD is elucidated with the help of one test illustration in which third order system is reduced to first order model as well as second order model. To prove applicability of the proposed method, impulse, step and Bode responses are plotted for both system and model. For relative comparison, time-domain specifications of proposed model are tabulated for both upper and lower limits. Further, performance indices are specified for dissimilarities between responses of system and model. The obtained results depict the effectiveness and efficacy for the proposed method.

Bounded-Input Bounded-Output Stability Tests for Two-Dimensional Continuous-Time Systems

This paper presents two efficient algorithms to determine whether a bivariate polynomial, possibly with complex coefficients, does not vanish in the cross product of two closed right-half planes (is “2-C stable”). A 2-C stable polynomial in the denominator of a two-dimensional analog filter has been proved (not long ago) to imply bounded-input bounded-output (BIBO) stability. The two algorithms are entirely different but both rely on a recently proposed fraction-free (FF) Routh test for complex polynomials in this transaction. The first algorithm tests the 2-C stability of a bivariate polynomial of degree (n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) in order n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> of elementary operations (when n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> =n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> =n). It is a “tabular type” two-dimensional stability test that can be regarded as a “Routh table” whose scalar entries were replaced by univariate polynomials. The second 2-C stability test is obtained from the first by its telepolation. It carries out the 2-C stability test by a finite collection of FF Routh tests and requires only order n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> elementary operations. Both algorithms possess an integer-preserving property that enhances them with additional merits including numerical error-free decision on 2-C stability.

Design of FOPID controller for higher order continuous interval system using improved approximation ensuring stability

This contribution deals with the design of a fractional-order proportional-integral-derivative (FOPID) controller through reduce-order modeling for continuous interval systems. First, a higher order interval plant (HOIP) is considered. The reduced-order interval plant (ROIP) for considered HOIP is derived by multipoint Padé approximation integrated with Routh table. Then, FOPID controller is designed for ROIP to satisfy the phase margin and gain cross over frequency. Thus obtained FOPID controller is implemented on HOIP also to validate the performance of designed FOPID on HOIP. A single-input-single-output (SISO) test system is taken up to elaborate the entire process of controller design. The outcomes affirm the validity of the designed FOPID controller. The designed FOPID controller produced stable results retaining the phase margin and gain cross-over frequency when implemented on HOIP. The results further proved that FOPID controller is working efficiently for ROIP and HOIP.

Anderson Corollary Based on New Approximation Method for Continuous Interval Systems

In this research, a new technique is developed for reducing the order of high-order continuous interval systems. The model denominator is derived using Anderson corollary and Routh table. Numerator is derived by matching the formulated Markov parameters (MPs) and time moments (TMs). Distinctive features of the proposed approach are: (i) New and simpler expressions for MPs and TMs; (ii) Retaining not only TMs but also MPs while deriving the model; (iii) Minimizing computational complexity while preserving the essential characteristics of system; (iv) Ensuring to produce a stable model for stable system; (v) No need to invert the system transfer function denominator while obtaining the TMs and MPs; and (vi) No need to solve a set of complex interval equations while deriving the model. Two single-input-single-output test cases are considered to illustrate the proposed technique. Comparative analysis is also presented based on the results obtained. The simplicity and effectiveness of the proposed technique are established from the simulation outcomes achieved.

Improved Approximation for SISO and MIMO Continuous Interval Systems Ensuring Stability

In the recent literature, some practical systems have been modeled as interval systems. In this investigation, an improved approximation technique is first proposed for model reduction in single-input-single-output continuous interval systems; then, the proposed technique is extended to reduce multi-input-multi-output continuous interval systems. The proposed technique is based on a multipoint Pade approximation. Additionally, the technique generates a stable model for a given stable system. The results show that the Routh table integrated multipoint Pade approximation is a good way to reduce continuous interval systems.

An improved method for reduction of continuous interval systems using Anderson corollary

Engineering systems are modeled as interval systems recently. In this paper, an improved technique for model reduction of high-order continuous interval systems is presented using Anderson corollary and Routh approximation. The distinguishing features of the proposed method are: (i) it ensures exact matching of responses of reduced-order interval model and high-order interval system at steady-state; (ii) it retains interval-value of steady-state of high-order interval system; and (iii) it produces interval model for interval system. Firstly, the denominator of the reduced-order model is obtained based on Routh table. Then, the numerator is obtained by matching the interval-values of steady-state of high-order interval system and reduced-order interval model. A single-input-single-output (SISO) test system is considered to explain the efficiency of the proposed method. The step responses of SISO test systems are obtained and presented for comparative study. From the results, it is observed that the proposed method provides better approximants.

Load Capacity Analysis of Self-Excited Induction Generators Based on Routh Criterion

In this paper, the Routh criterion has been used to analyze the stability of a self-excited induction generator-based isolated system which is regarded as an autonomous system. Special focus has been given to the load capacity of the self-excited induction generator. The state matrix of self-excited induction generators with resistor-inductor load has been established based on transient equivalent circuits in the stator stationary reference-frame. The recursive Routh table of self-excited induction generators is established by the characteristic polynomial coefficients of the state matrix. According to the Routh stability criterion, the necessary and sufficient condition to predict the critical loads of self-excited induction generators is deduced, from which the critical load impedance can be calculated. A simple self-excited induction generator-based isolated power system has been built up with a 2.2 kW self-excited induction generator. The theoretical analysis and experiments were all carried out based on this platform. In the range determined by the minimum excitation capacitance (Cmin) and the maximum excitation capacitance (Cmax), the critical loads under various power factors have been calculated. The agreement of the calculated theoretical results and experimental results demonstrate the effectiveness and accuracy of the proposed analysis method. The conclusions achieved lay a foundation for further application of Routh stability criterion in self-excited induction generator-based power systems analysis.

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.

Order Reduction in z-Domain for Interval System Using an Arithmetic Operator

This paper deals with order reduction in discrete-time systems. The reduction technique is based on utilization of Routh approximation and multiplicative operator. Various combinations of Routh table are used to derive the desired numerator and denominator polynomial of reduced-order model. Numerical examples are presented to verify the proposed algorithm.

Novel arrangement of Routh array for order reduction of z-domain uncertain system

ABSTRACTThis paper presents a novel arrangement of Routh table array for deriving an approximate model of a higher order z-domain uncertain system. The demand for this computation is to procure a lower order model which is easy to be exercised in comparison to their original large scale systems. Additionally, the derived model should preserve fewer dynamic characteristic of the comprehensive higher order systems. The mentioned new arrangement is achieved from the arena of different combinations of numerator and denominator polynomials. The combinations are validated by their practice over the conventional example from the literature. This precise blend is then applied to a real-time system for its rational acceptability. Both the models play a significant role in establishing the algorithm. Besides this, the limitation encountered during the foundation course of the arrangement is also taken into consideration. The paper also offers a future scope for fellow researchers.

Certain Algebraic Test for Analyzing Aperiodic Stability of Two-Dimensional Linear Discrete Systems

This paper addresses the new algebraic test to check the aperiodic stability of two dimensional linear time invariant discrete systems. Initially, the two dimensional characteristics equations are converted into equivalent one-dimensional equation. Further Fuller’s idea is applied on the equivalent one-dimensional characteristics equation. Then using the co-efficient of the characteristics equation, the routh table is formed to ascertain the aperiodic stability of the given two-dimensional linear discrete system. The illustrations were presented to show the applicability of the proposed technique.

Stable Routh-Padé-type approximation in model reduction of interval systems

An interval Pade-type approximation is introduced and then Routh-Pade-type method (IRPTM) is presented to model reduction in interval systems. The denominator in reduced model is obtained from the stable Routh table, and its numerator is constructed by the interval Pade-type definition. Compared to the existing Routh-Pade method, IRPTM does not need to solve linear interval equations. Hence, we do not have to compute interval division in the process. Moreover, theoretical analysis shows that IRPTM has smaller computational cost than that of Routh-Pade method. A typical numerical example is given to illustrate our method.

An interesting fact regarding the Routh table

The Routh table is a powerful analysis tool used to determine how many roots of a given polynomial have negative, zero, and positive real parts. A special case occurs when a row of zeros is encountered. This indicates an even polynomial may be factored from the given polynomial. It is well known that the even polynomial is in the row above the row of zeros. In this technical note it will be shown that the even polynomial may be factored from all of the polynomials in the rows above the row of zeros.

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

AbstractThe 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

Mixed method of model reduction for uncertain systems

A mixed method for reducing a higher order uncertain system to a stable reduced order one is proposed. Interval arithmetic is used to construct a generalized Routh table for determining the denominator polynomial of the reduced system. The reduced numerator polynomial is obtained using factor division method and the steady state error is minimized using gain correction factor. The proposed method is illustrated using a numerical example.

Inverse Routh table construction and stability of delay equations

Based on an inversion of the Routh table construction, a unimodular characterization of all Hurwitz polynomials is obtained. In parameter space, the Hurwitz polynomials of degree n correspond to the positive 2 n - tant . The method is then used to construct classes of stable continuous-time delay-difference equations and delay differential equations of neutral type, by a suitable limiting process.

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.

On routh-pade model reduction of interval systems

This paper discusses another generalization of the direct Routh table truncation method for interval systems. It is shown that the existing generalization of the direct Routh table truncation fails to produce a stable system, in contradiction to the equivalent result for fixed-coefficients systems. The present method guarantees a stable reduced order model for interval systems as well.

What Can Routh Table Offer in Addition to Stability?

Routh stability test is covered in almost all undergraduate control text. It determines the stability or, a little 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 Routh table, as far as the common textbooks show, is only limited in this function. In this paper, we will show that Routh table can actually 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.