Routh Stability Research Articles

An application of Routh stability criterion to derive the necessary conditions for multiplicity of a CSTR

AbstractIt is well known that the Routh stability criterion (RSC) is a powerful method for analyzing control system stability. A new RSC finding that can aid the tangent analysis method (TAM) to determine the necessary conditions for multiplicity is presented. Adding RSC to TAM can find the bifurcation start point which cannot be obtained using TAM alone. To the author's knowledge, the above new RSC application has not been discussed in past literatures. Our results are helpful for chemical reactor design. We used two examples of the binary reaction in a non‐adiabatic CSTR to demonstrate the merit of our methods.

Relationship on Stabilizability of LTI Systems by P and PI Controllers

AbstractThis paper investigates the relationship on stabilizability of linear time‐invariant (LTI) systems by P and PI controllers. Elementary tools such as the Routh stability criterion and Root‐Locus method are employed in the analysis. It is found that PI can stabilize all the systems that P stabilizes but in general the converse is not true. The cases with the equivalence of stabilizability by P and PI are established and they are in general low‐order systems with few zeros. The cases with non‐equivalence are also identified.

A theoretic control approach in signal-controlled metabolic pathways

Cells use a signal transduction mechanism to regulate certain metabolic pathways. In this paper, the regulatory mechanism is analyzed mathematically. For this analysis, a mathematical model for the pathways is first established using a system of differential equations. Then the linear stability, controllability, and observability of the system are investigated. We show that the linearized system is controllable and observable, and that the real parts of all eigenvalues of the linearized system are nonpositive using Routh's stability criterion. Controllability and observability are structural properties of a dynamical system. Thus our results may explain why the metabolic path ways can be controlled and regulated. Finally observer-based and proportional output feedback controllers are designed to regulate the end product to its de sired level. Applications to the regulation of blood glucose levels are discussed.

Analytic Expressions of Ultimate Gains and Ultimate Periods with Phase‐Optimal Approximations of Time Delays

AbstractUltimate gains and ultimate periods have long been used to tune PI/PID controllers since the Ziegler‐Nichols tuning. Recently, very simple analytic equations for ultimate gains and ultimate periods by approximating the arctangent function have been presented, which can be used to obtain simple tuning rules. Here, by applying the Routh stability method and phase‐optimal approximations of the time delay term, we present analytic expressions of ultimate parameters that are as simple as the existing ones, showing better accuracy.

Dynamics of multibody systems in planar motion in a central gravitational field

Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations to these relative equilibria, to confirm their stability properties.

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.

Control Routh Array And Its Applications

ABSTRACTIn 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.

A stable series expansion method for calculating thermal response factors of multi-layer walls

Thermal response factors (TRFs) of walls are fundamental data for calculating air conditioning loads and annual energy consumption of buildings. Among the methods for calculating TRFs, the state space method developed by Ouyang and Haghighat (Build. Environ. 26(2)(1991)173) is based on the s-transfer functions but has the advantage of avoiding conventional root-finding process. However, it is found that the calculation of TRFs may not be convergent with increase of the terms in expansion polynomials. In this paper, detailed analysis shows that the problem originates from the instability of reduced s-transfer functions obtained from the power series expansion. As a substitute, a stable series expansion approach based on the Routh stability theory is proposed. It keeps advantages of the existing method and guarantees the stability in calculation of TRFs.

Stability and performance analysis in an uncertain world

Classical control systems design uses fixed plant transfer functions, yet engineers have known for many years that there is often considerable uncertainty regarding the parameters used in a transfer function representation. A major breakthrough on systems with uncertain parameters was achieved by the Russian mathematician V.L. Kharitonov (1979), who extended the Routh stability criterion to interval polynomials, that is where the polynomial coefficients may be assumed to lie within a specific range rather than being fixed. Recently these ideas have been extended to frequency response representations, such as Nyquist and Bode plots, for interval plant transfer functions. This paper reviews some of these results, which can be easily understood and used by practising engineers.

Process dynamics: modeling, analysis, and simulation

I. PROCESS MODELING. 1. Introduction. Motivation. Models. Systems. Background of the Reader. How To Use This Textbook. Courses Where This Textbook Can Be Used. 2. Process Modeling. Background. Balance Equations. Material Balances. Constitutive Relationships. Material and Energy Balances. Distributes Parameter Systems. Dimensionless Models. Explicit Solutions to Dynamic Models. General Form of Dynamic Models. II. NUMERICAL TECHNIQUES. 3. Algebraic Equations. Notations. General Form for a Linear System of Equations. Nonlinear Functions of a Single Variable. MATLAB Routines for Solving Functions of a Single Variable. Multivariable Systems. MATLAB Routines for Systems of Nonlinear Algebraic Equations. 4. Numerical Integration. Background. Euler Integration. Runge-Kutta Integration. MATLAB Integration Routines. III. LINEAR SYSTEMS ANALYSIS. 5. Linearization of Nonlinear Models: The State-Space Formulation. State Space Models. Linearization of Nonlinear Models. Interpretation of Linearization. Solution of the Zero-Input Form. Solution of the General State-Space Form. MATLAB Routines step and initial. 6. Solving Linear nth Order ODE Models. Background. Solving Homogeneous, Linear ODEs with Constant Coefficients. Solving Nonhomogeneous, Linear ODEs with Constant Coefficients. Equations with Time-Varying Parameters. Routh Stability Criterion-Determining Stability Without Calculating Eigenvalues. 7. An Introduction to Laplace Transforms. Motivation. Definition of the Laplace Transform. Examples of Laplace Transforms. Final and Initial Value Theorems. Application Examples. Table of Laplace Transforms. 8. Transfer Function Analysis of First-Order Systems. Perspective. Responses of First-Order Systems. Examples of Self-Regulating Processes. Integrating Processes. Lead-Lag Models. 9. Transfer Function Analysis of Higher-Order Systems. Responses of Second-Order Systems. Second-Order Systems with Numerator Dynamics. The Effect of Pole-Zero Locations on System Step Responses. Pad Approximation for Deadtime. Converting the Transfer Function Model to State-Space Form. MATLAB Routines for Step and Impulse Response. 10. Matrix Transfer Functions. A Second-Order Example. The General Method. MATLAB Routine ss2tf. 11. Block Diagrams. Introduction to Block Diagrams. Block Diagrams of Systems in Series. Pole-Zero Cancellation. Systems in Series. Blocks in Parallel. Feedback and Recycle Systems. Routh Stability Criterion Applied to Transfer Functions. SIMULINK. 12. Linear Systems Summary. Background. Linear Boundary Value Problems. Review of Methods for Linear Initial Value Problems. Introduction to Discrete-Time Models. Parameter Estimation of Discrete Linear Systems. IV. NONLINEAR SYSTEMS ANALYSIS. 13. Phase-Plane Analysis. Background. Linear System Examples. Generalization of Phase-Plane Behavior. Nonlinear Systems. 14. Introduction Nonlinear Dynamics: A Case Study of the Quadratic Map. Background. A Simple Population Growth Model. A More Realistic Population Model. Cobweb Diagrams. Bifurcation and Orbit Diagrams. Stability of Fixed-Point Solutions. Cascade of Period-Doublings. Further Comments on Chaotic Behavior. 15. Bifurcation Behavior of Single ODE Systems. Motivation. Illustration of Bifurcation Behavior. Types of Bifurcations. 16. Bifurcation Behavior of Two-State Systems. Background. Single-Dimensional Bifurcations in the Phase-Plane. Limit Cycle Behavior. The Hopf Bifurcation. 17. Introduction to Chaos: The Lorenz Equations. Introduction. Background. The Lorenz Equations. Stability Analysis of the Lorenz Equations. Numerical Study of the Lorenz Equations. Chaos in Chemical Systems. Other Issues in Chaos. IV. REVIEW AND LEARNING MODULES. Module 1 Introduction to MATLAB. Module 2 Review of Matrix Algebra. Module 3 Linear Regression. Module 4 Introduction to SIMULINK. Module 5 Stirred Tank Heaters. Module 6 Absorption. Module 7 Isothermal Continuous Stirred Tank Chemical Reactors. Module 8 Biochemical Reactors. Module 9 Diabatic Continuous Stirred Tank Reactors. Module 10 Ideal Binary Distillation. Index.

Hgram patterns of Routh stability zones in linear systems

For a generalized nth‐orderlinear system, the characteristic equation can be expressed in a normalized form a 0l • sn + a 1 sn‐ 1 + a 2 sn‐2 + ... + an‐ 1 sn‐ 1 +an = 0.There are thus nindependent coefficients to be adjusted to achieve system stability. The stability zones can be mapped by scanning the coefficients a 1 to an in multi‐dimensional Hgram graphical space using Routh stability criterion. Two methods of multidimensional display are considered, namely the ‘circular walk’ or continuous mode and the grid or discrete mode. The colours of the linklines can be toggled to distinguish between stability and instability determined by Routh stability tests. Unique patterns called Hgram patterns which characterize Routh stability were obtained for linear systems of the 4th and 6th orders. Besides providing better insight on the stability characteristics of high‐order linear systems, graphical results are mnemonically recallable and should be welcomed by those engaged in training, research and design. Frequently, CAH geometers prefer to use a hybrid display mode taking advantages of both the discrete and the continuous modes.

A new two-step iterative method for optimal reduction of linear SISO systems

A new two-step iterative procedure is proposed for the optimal reduced-order modeling of linear time-invariant single-input single-output (SISO) systems. The performance index of optimal reduction is taken to be a quadratic function of the error between the time responses of the original and reduced models. At each iteration cycle, the numerator dynamics is first determined by solving a set of linear equations, and the denominator polynomial is then determined by a gradient-based search technique. The main features of the proposed procedure are that it searches the Routh stability parameters rather than the denominator polynomial coefficients of the reduced model, and computes the performance index and its gradients by a computationally efficient parametric algorithm. As a consequence, the need of stability monitoring in the step of searching optimal denominator polynomial for the reduced model is avoided, and the gradient vector evaluated exactly and efficiently for a gradient-based parameter search. Moreover, the constraint of zero steady-state response error can be easily handled.

Dynamic elastic instability of long circular cylindrical shells under pure bending

The present paper studies the responses and instabilities of long circular cylindrical shells subjected to dynamic pure bending. The dynamic instability characteristics of the shells subjected to a sudden step bending load of infinite duration are explored. Analysis is performed using nonlinear finite element numerical methods. Critical dynamic moments are determined through the use of Budiansky and Routh's stability criterion. Numerical predictions for the dynamic instability are compared with those static results given earlier by Brazier. The effects of shell geometry on the dynamic stability of the shells are shown. It is found that the dominating factor to affect the shell stability is the ovalization of the shell cross-section in the centre of the shell.

Computation of optimal reduced-order models with time delay

This paper considers the problem of optimally approximating high-order transfer functions by low-order ones with a time delay. The performance index used for optimization is the integral of squared error (ISE) between the unit-step responses of the reduced-order and original transfer functions. To facilitate using gradient-based methods to search optimal model parameters, an effective numerical method is adapted for evaluating the ISE and its gradient with respect to parameters. The method is based on using Parseval's identify and the variable transformation ▪, such that the accurate evaluation of ISE involving time delay is performed via solving a first-order differential equation. In determining the optimal denominator polynomial for the reduced-order transfer function, the Routh stability parameters rather than the polynomial coefficients are searched. Such a substitution avoids the effort of stability check at each iteration of parameter search.

Root distribution of a polynomial in subregions of complex plane

Modified processes are proposed for directly treating singularities in the Routh array, and procedures are developed for determining the respective orders of simple and/or repeated roots lying on the imaginary axis. The extended Sturm test is developed for determining the root distribution on some specified lines. Moreover, to enhance the Routh stability theorem, the extended Routh theorem for finding the numbers of roots of a polynomial lying within some specified regions is proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Adaptive Position Control of an Electro Hydraulic Servo Cylinder.

This paper is concerned with the practical application of model reference adaptive control (MRAC) on the position control of a hydraulic cylinder. In this study, the modified hyperstable-MRAC method is employed to adapt the loading effect. The reference model is based on the reduced-order model of the plant, which is derived by using the Routh stability criterion and the Pade approximation technique. A 16-bit microcomputer is used to design an adaptive controller. The experimental results are also compared with those obtained by using the classical control theory.

Phase margin of linear time invariant systems from Routh array

A simple method to determine the gain margin of linear time invariant systems (LTIS) employing Routh stability criterion was developed in an earlier paper. The term 'stability ratio' was defined connecting the marginal gain K/sub m/ determined from the Routh array and the actual gain K of the system. This established a direct link between the Routh criterion and the gain margin. It was also shown that the phase cross-over frequency omega /sub cp/ can be found from the auxiliary equation formed from the elements of the s/sup 2/-row of the Routh array. Empirical formulae for computing the gain cross-over frequency omega /sub cg/ of type 0 systems and of higher-type systems in terms of system gain K and the marginal gain K/sub m/ are suggested. The gain cross-over frequency and phase margin calculated using the empirical formulae are in close agreement with exact values, justifying the use of the empirical formulae. It is shown that the Routh stability criterion can be used to measure the relative stability of LTIS.

Control systems engineering and design

Control system design - Lagrange's equations differential equations and Laplace transforms block diagrams design concepts in the s-plane - graphical interpretation of Heaviside's partial fraction stability - Routh stability and s-plane performance criteria, Routh array and PID controller design, Ziegler-Nichols rules for controller tuning root loci frequency response design methods - Bode plots, polar plots, the Nyquist contour controller and compensator design identification - Bodes decomposition method, Levy's method case studies - feedback process trainer PT326, Armfield PCT flow control experiment, techquipment CE8 coupled electric drives design and identification programmes - a CACSD package, numerical integration - the Euler-Cauchy method, Horner's method, transfer function identification - Levy's method.

Hybrid method for model reduction with time delay

A novel approach to the simplification of dynamic systems is introduced. It is suggested that a high-order linear system can be approximated by a low-order model with a time delay based on the proposed mixed method. The denominator of the reduced model is determined by the Routh stability method, which gives rise to the desired Stability of the model produced. The numerator parameters are then estimated by using identification techniques and the time delay is chosen optimally according to an output-errors criterion. It is shown that the reduced-order time-delay model produces a good enough approximation compared with models obtained from other methods. The reduction procedure is fully computer oriented. The performance of the new approach is verified by the application of numerical examples.

Transputer implementation of systolic arrays for model reduction

In the paper the author develops a unified bidirectional linear systolic array for performing model reduction of a feedback control system using continued fractions. The array is capable of computing the 1st, 2nd and 3rd Cauer forms of the fraction and their associated inverses as well as the Routh stability criterion. Timing graphs are presented to indicate the speed-up obtained on five transputers for common problem ranges.