Schur-Cohn Stability Research Articles

Modifications of the Samuelson Economic Cycle Model

Abstract The paper presents an analysis of modifications of the standard discrete Samuelson model with determination of the stability of the equilibrium point. The stability region of the equilibrium point depending on the parameters contained in each model is determined using the Schur-Cohn stability criterion.

Symmetric Properties of Routh–Hurwitz and Schur–Cohn Stability Criteria

It is often noticed in the literature that some key results on the stability of discrete-time systems of difference equations are obtained from their corresponding results on the stability of continuous-time systems of differential equations using suitable conformal mappings or bilinear transformations. Such observations lead to the search for a unified approach to the study of root distribution for real and complex polynomials, with respect to the left-half plane for continuous-time systems (Routh–Hurwitz stability) and with respect to the unit disc for discrete-time systems (Schur–Cohn stability). This paper is a further contribution toward this objective. We present, in a systematic way, the similarities, and yet, the differences between these two types of stability, and we highlight the symmetry that exists between them. We also illustrate how results on the stability of continuous-time systems are conveyed to the stability of discrete-time systems through the proposed techniques. It should be mentioned that the results on Schur–Cohn stability are known to be harder to obtain than Routh–Hurwitz stability ones, giving more credibility to the proposed approach.

Structural properties of a class of functions arising in stability theory

Functions like positive para-odd and complex discrete reactance functions play important roles in both Routh-Hurwitz and Schur-Cohn stability theories. Other functions like circular symmetric and circular antisymmetric have impacted stability problems in various ways. It is the objective of this paper to highlight some of the important structural properties of these functions and relations between them, leading to new characterizations of antisymmetric and complex discrete reactance functions both in terms of circular symmetric and circular antisymmetric functions.

Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal $$\mathbb {C}\times \mathbb {S}^1$$ -bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic $$\mathbb {C}$$ -actions $$\mathscr {A}$$ on the space of polynomials of degree n. For each orbit $$\{ s \cdot P(z) \ \vert \ s \in \mathbb {C}\}$$ of $$\mathscr {A}$$ , we study the dynamical problem of the existence of a complex rational vector field $$\mathbb {X}(z)$$ on $$\mathbb {C}$$ such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit $$\{ s \cdot P(z) = 0 \}$$ . Regarding the above $$\mathbb {C}$$ -action coming from the $$\mathbb {C}\times \mathbb {S}^1$$ -bundle structure, we prove the existence of a complex rational vector field $$\mathbb {X}(z)$$ on $$\mathbb {C}$$ , which describes the geometric change of the n-root configuration in the unitary disk $$\mathbb {D}$$ of a $$\mathbb {C}$$ -orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in $$\mathbb {C}\backslash \overline{\mathbb {D}}$$ , by constructing a principal $$\mathbb {C}^* \times \mathbb {S}^1$$ -bundle structure in this family of polynomials. As an application for a cohort population model, a study of the Schur stability and a criterion of the loss of Schur stability are described.

New Versions of the Hermite Bieler Theorem in Stability Contexts

The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equations. Using the theory of positive paraodd functions, we obtain Hermite-Bieler like conditions for the Routh-Hurwitz stability of such systems. We also look at the problem of stability of discrete-time systems of difference equations. By using suitable conformal mappings, we also establish Hermite-Bieler like conditions for the Schur-Cohn stability of these systems. In both cases, the conditions are necessary as well as sufficient.

The Probability of Stability Estimation of an Arbitrary Order DPCM Prediction Filter: Comparison between the Classical Approach and the Monte Carlo Method

This paper presents the stability analysis of the linear recursive (prediction) filters with higher order predictors in a DPCM (differential pulse-code modulation) system, where traditional methods become too difficult and complex. Stability conditions for the third- and fourth-order predictor are given by using the Schur–Cohn stability criterion. The probability of stability estimation is performed by using the Monte Carlo method. Verification of the proposed method is performed for lower order predictors (the first- and second-order). We calculated numerical values of the probability of stability for higher order predictors and previously experimentally obtained parameters. With large enough number of trials (samples) in Monte Carlo simulation, we reach the desired accuracy. DOI: http://dx.doi.org/10.5755/j01.itc.46.2.14038

On Schur-Cohn stability of a product of operators in a Hilbert space

On Schur-Cohn stability of a product of operators in a Hilbert space

ODMCA: An adaptive data mining control algorithm in multicarrier networks

Multicarrier communication is a promising technique to effectively deliver high data rate and combat delay spread over fading channel, and adaptability is an inherent advantage of multicarrier communication systems. It can be implemented in online data streams. This paper addresses a significant problem in multicarrier networks that arises in data streaming scenarios, namely, today’s data mining is ill-equipped to handle data streams effectively, and pays little attention to the network stability and the fast response [ http://www-db.standford.edu/stream]. Furthermore, in analysis of massive data streams, the ability to process the data in a single pass, while using little memory, is crucial. For often the data can be transmitted faster than it can be stored or accessed from disks. To address the question, we present an adaptive control-theoretic explicit rate (ER) online data mining control algorithm (ODMCA) to regulate the sending rate of mined data, which accounts for the main memory occupancies of terminal nodes. This single-pass scheme considers limited memory space to process dynamic data streams, and also explores the adaptive capability, which is employed in a general network computation model for dynamic data streams. The proposed method uses a distributed proportional integrative plus derivative (PID) controller combined with data mining, where the control parameters can be designed to ensure the stability of the control loop in terms of sending rate of mined data. The basic PID approach for the computation network transmission is presented and z-transformation and Schur–Cohn stability test are used to achieve the stability criterion, which ensures the bounded rate allocation without steady state oscillation. We further show how the ODMCA can be used to design a controller, analyze the theoretical aspects of the proposed algorithm and verify its agreement with the simulations in the LAN case and the WAN case. Simulation results show the efficiency of our scheme in terms of high main memory occupancy, fast response of the main memory occupancy and of the controlled sending rates.

Direct Form I Realization of Active Photonic Filters

An integrated photonic architecture is introduced and used to realize an optical filter with direct form I realization. The architecture offers gain from semiconductor optical amplifiers, and this gain results in an active optical filter whose filter response depends on the individual gains. The presence of gain provides advantages in filter performance, and tunable and adaptive functionality. The optical filter is modeled as a discrete time system and the z-transform is used in its analysis and design. A low-pass filter design example is presented and the filter coefficients are derived in terms of gains and coupler splitting ratios. The region of stable operations is derived by applying the Schur-Cohn stability test.

Multidimensional Schur coefficients and BIBO stability

In the framework of BIBO stability tests for one-dimensional (1-D) linear systems, the Schur-Cohn stability test has the appealing property of being a recursive algorithm. This is a consequence of the simultaneously algebric and analytic aspect of the Schur coefficients, which can be also regarded as reflection coefficients. In the multidimensional setting, this dual aspect gives rise to two extension of the Schur coefficients that are no longer equivalent. This paper presents the two extensions of the Schur-Cohn stability test that derive from these extended Schur coefficients. The reflection-coefficient approach was recently proposed in the 2-D case as a necessary but non sufficient condition of stability. The Schur-type multidimensional approach provides a stronger condition of stability, which is necessary and sufficient condition of stability for multidimensional linear system. This extension is based on so-called slice function associated to n-variable analytic functions. Several examples are given to illustrate this approach.

Extension of the schur-cohn stability test for 2-d ar quarter-plane model

The Schur-Cohn test plays an essential role in checking the stability of one-dimensional (1D) random processes such as autoregressive (AR) models, via the so-called reflection coefficients, partial correlations, or Schur-Szego coefficients. In the context of two-dimensional (2D) random field modeling, one of the authors recently proposed a 2D AR quarter-plane model representation using 2D reflection coefficients estimated by a fast recursive adaptive algorithm. Based on such 2D reflection coefficients, we can therefore derive two necessary stability conditions for a 2D AR quarter-plane model. One of these conditions can be considered as an extension of the Schur-Cohn stability criterion based on the 2D reflection coefficients.

Rate-based congestion control in ATM switching networks using a recursive digital filter

The present paper proposes a control-theoretic approach to design rate-based controllers in order to flow-regulate the best-effort service in asynchronous transfer mode (ATM) switching networks. The proposed control uses a recursive digital filtering controller, where the control parameters can be designed to ensure the stability of the control loop in a control-theoretic sense. The stability of closed-loop congestion-controlled systems is analyzed by using Schur–Cohn stability criterion, which leads to certain necessary and sufficient stability condition under which the controlled ATM switching network is asymptotically stable in terms of buffer occupancy. Such proposed stability condition is then shown to be a key tool in designing a wide scope of adaptive controllers. Further, we demonstrate that a fair share of the available bandwidth at the bottleneck node can be achieved according to the proposed control policy. Simulations are performed that show good performance of both local area networks and wide area networks if implemented by the proposed control schemes.

RATE-BASED CONGESTION CONTROLLERS FOR HIGH-SPEED COMPUTER NETWORKS

The present paper proposes a control-theoretic approach to design rate-based controllers in order to flow-regulate the best-effort traffic through high-speed computer communication networks. Classical control theory and Schur-Cohn stability test are exploited to design the traffic controllers for high-speed networks. The stability of closed-loop congestion controlled systems is analysed by utilizing Schur-Cohn stability criterion, which leads to certain necessary and sufficient stability condition under which the controlled network is asymptotically stable in terms of buffer occupancy. The proposed stability condition is then shown to be a key tool in designing a wide scope of adaptive controllers. Simulations are performed that show good performance of such controlled networks.

A normalized Schur-Cohn stability test for the delta-operator-based polynomials

The author previously (1997) proposed two delta-operator-based stability tests, or more generally zero location tests. Those tests establish two families of such tests, each spanning from the discrete-time to the continuous-time with the delta operator providing smooth transitions between the two domains. In this paper a third family is proposed. Specifically, the normalized Schur-Cohn test in the discrete-time domain is transformed into the delta-operator domain resulting in a new delta-operator test. The limit of this new test as the sampling interval vanishes is shown to be the Pham-Le Breton test (1991) in the continuous-time domain. Its relationships with the well-known Routh test and others are studied. A numerical example shows the advantage of the new test for fast sampling.

Computation of the stability margin of two-dimensional discrete systems

Two methods of computing the stability margin of two-dimensional discrete systems are presented. The first method uses the Schur-Cohn stability criterion, and the second method is based on results concerning the 1-D Lyapunov equation with complex coefficients. Several examples illustrate the applicability of the two methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A useful statement of some Schur-Cohn stability criteria for higher order discrete dynamic systems

The classic Schur-Cohn criteria are an important source of stability conditions for discrete dynamic systems. However, the conventional statement of these criteria is noted to be opaque and not of much direct use for an applied analysis of higher-order dynamic systems. In the present work, the author shows how some of these criteria can be stated as useful and easily tested restrictions on the gradient matrices of the original higher order system. >

On another approach to the Schur-Cohn criterion

Some well-known properties of polynomials orthogonal on the unit circle are used to provide a simple proof of the Schur-Cohn stability criterion. Some complements are also briefly noted.

On the singularity of influence coefficients of the externally pressurized spherical shells

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