Unstable Characteristic Roots Research Articles

An algorithm to design a stabilizing feedback controller for linear delay systems via Parseval ’s theorem

We investigate feedback stabilization problems of a linear delay system. When the unstable characteristic roots of the system are far from the imaginary axis, the discretization of the system results in a large error. In this case, it is difficult to seek stabilizing control laws via the algorithm by Hu and Hu, (2020). To avoid the discretization of unstable differential equations, a modified state equation is constructed through a shifting parameter such that the equation is asymptotically stable. Then, based on the modified state equation and Parseval’s theorem, a numerical optimization algorithm is provided to design a stabilizing controller. Meanwhile, we compare the presented algorithm with those in the literature. Finally, numerical examples show that the presented algorithm in this paper is efficient.

A Current Sensorless Delay–Based Control Scheme for MPPT–Boost Converters in Photovoltaic Systems

The results presented in this paper deal with the design of a current sensorless delay–based controller for the closed–loop stabilization of a photovoltaic system under an MPPT scheme using a boost dc/dc converter. Some applications of such topology are dc microgrids, solar vehicles, or stand-alone systems, to mention a few. The basis of this control scheme relies on the feedback linearization control technique coupled with a delay–based low-order controller. In order to study the stability, the proposed approach uses a geometric point of view which allows the partitioning of the controller parameters space into regions with similar stability characteristics (same number of unstable characteristic roots). The most important contribution of the paper relies on providing practical guidelines to tune the gains of the proposed delay–based controller, ensuring asymptotic stability of the closed–loop system and fulfilling the requirements for photovoltaic applications. In addition, the proposed approach allows the design a non-fragile controller with respect to the controller gains. Furthermore, in order to test the effectiveness of the control scheme presented, experimental results evaluating the closed–loop system performance under set-point changes and abrupt irradiance disturbances are addressed using a solar array simulator and a battery bank as load.

Stability criteria for neutral delay differential-algebraic equations with many delays

In this paper the asymptotic stability is concerned for a class of neutral delay differential-algebraic equations (NDDAEs). We will present two criteria by evaluating a corresponding harmonic function on the boundary of a torus region. Stability regions are also presented so as to locate all possible unstable characteristic roots of NDDAEs. As we know, these roots will make bad numerical simulations. Our criteria help find and avoid them. Numerical examples are shown to check our criteria.

Reversals in stability of linear time-delay systems: A finer characterization

In most of the numerical examples of time-delay systems proposed in the literature, the number of unstable characteristic roots remains positive before and after a multiple critical imaginary root (CIR) appears (as the delay, seen as a parameter, increases). This fact may lead to some misunderstandings: (i) A multiple CIR may at most affect the instability degree; (ii) It cannot cause any stability reversals (stability transitions from instability to stability). As far as we know, whether the appearance of a multiple CIR can induce stability is still unclear (in fact, when a CIR generates a stability reversal has not been specifically investigated). In this paper, we provide a finer analysis of stability reversals and some new insights into the classification: the link between the multiplicity of a CIR and the asymptotic behavior with the stabilizing effect. Based on these results, we present an example illustrating that a multiple CIR’s asymptotic behavior is able to cause a stability reversal. To the best of the authors’ knowledge, such an example is a novelty in the literature on time-delay systems.

Retarded, neutral and advanced differential equation models for balancing using an accelerometer

Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the modeling concepts, the governing equations can be a retarded functional differential equation or neutral functional differential equation or even advanced functional differential equation. These types of equations have radically different stability properties. In the retarded and the neutral case the system can be stabilized, but the advanced equations are always unstable with infinitely many unstable characteristic roots. It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models. It is concluded that digital effects, such as sampling, stabilizes the system independently on the modeling details.

Stability and delay sensitivity of neutral fractional-delay systems.

This paper generalizes the stability test method via integral estimation for integer-order neutral time-delay systems to neutral fractional-delay systems. The key step in stability test is the calculation of the number of unstable characteristic roots that is described by a definite integral over an interval from zero to a sufficient large upper limit. Algorithms for correctly estimating the upper limits of the integral are given in two concise ways, parameter dependent or independent. A special feature of the proposed method is that it judges the stability of fractional-delay systems simply by using rough integral estimation. Meanwhile, the paper shows that for some neutral fractional-delay systems, the stability is extremely sensitive to the change of time delays. Examples are given for demonstrating the proposed method as well as the delay sensitivity.

Decomposition of an autoregressive process into first order processes

Let Yn be an autoregressive process of order p. With p distinct characteristic roots, Yn can be decomposed into or expressed as a linear combination of p first order autoregressive processes. For the case of multiple characteristic roots, Yn with s<p distinct characteristic roots can be expressed as a linear combination of s first order autoregressive processes and the derivatives with respect to the parameter of the s first order processes. The parameters of the first order processes are the characteristic roots of Yn. Using this decomposition, for general stationary and unstable characteristic roots, the limiting distribution of appropriately normalized maximum likelihood estimators for the parameters of Yn are obtained. These results are new to the literature.

Computable stability criterion of linear neutral systems with unstable difference operators

This paper is concerned with linear neutral systems with unstable difference operators. All the unstable characteristic roots of the linear neutral systems are located in a bounded half circular region. Then, based on the argument principle, a computable stability criterion is derived which is necessary and sufficient for asymptotic stability of the neutral systems. Furthermore, the number of the unstable characteristic roots is given. The main results of the present paper extend those in literature. Numerical experiments are provided to illustrate the main results.

Computation of Unstable Characteristic Roots of Neutral Delay Systems

摘要: 提出了一种基于幅角原理计算一类中立型延时系统的所有不稳定特征根的算法. 通过将右半复平面内的有界矩形区域或半圆形区域连续划分成较小的区域, 能够有效地获得所有不稳定特征根的初始近似位置. 以这些近似位置作为牛顿法的初始值, 可迭代得到所有不稳定特征根的较好近似值.数值算例显示了算法的有效性. 关键词: 中立型延时系统 / 特征根 / 幅角原理 / 寻根

Computation of Unstable Characteristic Roots of Neutral Delay Systems

We present an algorithm based on the argument principle for computing all unstable characteristic roots of a class of neutral time delay systems. By consecutively subdividing a bounded rectangular or half circular region on the right half complex plane into smaller ones, initial approximate positions of all unstable roots can be located efficiently. With these approximate positions as starting points for Newton's method, approximations for all roots are refined iteratively. The performance of the algorithm is shown by an example.

On the geometry of pi controllers for SISO systems with input delays

This paper focuses on the stability analysis of the closed-loop single-input-single-output (SISO) system subject to input delays and to PI controllers. A complete characterization of the crossing set, which consists of all frequencies where the number of unstable characteristic roots changes, is given in the parameter space defined by the controller. Next, we explicitly derive the PI controllers corresponding to each frequency in the crossing set. More precisely, we make a partition of the controller parameter space in regions where the number of unstable characteristic roots remains constant. Finally, we also present a methodology to compute the number of unstable roots in each region. Several illustrative examples end the presentation.

An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type

An eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented, which extends the recently developed continuous pole placement method for delay equations of retarded type. The approach consists of two steps. First the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots, which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation. No restrictions are made on the dimension of the system and the number of delays. Also the effect of small delay perturbations is explicitly taken into account. As a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots. As a second step, stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi-continuous way, by applying small changes to the controller parameters, and meanwhile monitoring other characteristic roots with a large real part. A numerical example is presented.

The Cluster Treatment of Characteristic Roots and the Neutral Type Time-Delayed Systems

A new methodology is presented for assessing the stability posture of a general class of linear time-invariant—neutral time-delayed systems (LTI-NTDS). It is based on a “Cluster Treatment of Characteristic Roots CTCR” paradigm, which yields a procedure called the Direct Method (DM). The technique offers a number of unique features: It returns exact bounds of time delay for stability, as well as the number of unstable characteristic roots of the system in an explicit and nonsequentially evaluated function of time delay. As a direct consequence of the latter feature, the new methodology creates entirely, all existing stability intervals of delay, τ. It is shown that the Direct Method inherently enforces an intriguing necessary condition for τ-stabilizability, which is the main contribution of this paper. This, so-called, “small delay” effect, was recognized earlier for NTDS, only through some cumbersome mathematics. Furthermore, the Direct Method is also unique in handling systems with unstable starting posture for τ=0, which may be τ-stabilized for higher values of delay. Example cases are provided.

A practical method for analyzing the stability of neutral type LTI-time delayed systems

A new paradigm is presented for assessing the stability posture of a general class of linear time invariant–neutral time delayed systems ( LTI-NTDS). The ensuing method, which we name the direct method ( DM), offers several unique features: It returns the number of unstable characteristic roots of the system in an explicit and non-sequentially evaluated function of time delay, τ. Consequently, the direct method creates exclusively all possible stability intervals of τ. Furthermore, it is shown that this method inherently verifies a widely accepted necessary condition for the τ-stabilizability of a LTI-NTDS. In the core of the DM lie a root clustering paradigm and the strength of Rekasius transformation in mapping a transcendental characteristic equation into an equivalent rational polynomial. In addition, we also demonstrate by an example that DM can tackle systems with unstable starting posture for τ=0, only to stabilize for higher values of delay, which is rather unique in the literature.

Direct method for analyzing the stability of neutral type LTI-time delayed systems

A new paradigm is presented for assessing the stability posture of a general class of linear time invariant - neutral time delayed systems (LTI-NTDS). Unlike peer techniques, this novel method, called the Direct Method, offers a number of unique features: It returns exact bounds of time delay for stability including separated stability pockets: beyond that it gives the number of unstable characteristic roots of the system in an explicit function of time delay, τ. As a direct consequence of the latter feature, it creates exclusively all possible stability intervals of τ. Furthermore it is shown that the Direct Method inherently enforces a very popular necessary condition for the stabilizability of LTI-NTDS. In the core of the approach lies the strength of Rekasius transformation, which maps (exactly) the transcendental characteristic equation of LTI-NTDS into an equivalent rational polynomial form. In addition to the above listed unmatched characteristics of the Direct Method we also demonstrate that it can tackle systems with unstable starting posture for τ = 0, only to stabilize it for higher values of delay, which is rather unique in the literature.

An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems

A general class of linear time invariant systems with time delay is studied. Recently, they attracted considerable interest in the systems and control community. The complexity arises due to the exponential type transcendental terms in their characteristic equation. The transcendentality brings infinitely many characteristic roots, which are cumbersome to elaborate as evident from the literature. A number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay. This study offers an exact, structured and robust methodology to bring a closure to the question at hand. Ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions (pockets) in the domain of time delay, but it also declares the number of unstable characteristic roots at any given pocket. The method starts with the determination of all possible purely imaginary (resonant) characteristic roots for any positive time delay. To achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation. It is proven that the number of such resonant roots for a given dynamics is finite. Each one of these roots is created by infinitely many time delays, which are periodically distributed. An interesting property is also claimed next, that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency. These two unique findings facilitate a simple and practical stability method, which is the highlight of the work.

Robust stability for noncommensurate time-delay systems

Based on the maximum principle and some properties of matrix measure and matrix norm, several new stability criteria for perturbed multiple noncommensurate time-delays systems are proposed. These criteria ensure all roots of the system's characteristic equation lying on or to the left of Re(s)=-/spl alpha/, /spl alpha/>o. Two approaches are used here, one is a direct test which needs some matrix computations and the other is a loci test in which one needs to draw the loci of eigenvalues. With the aid of the convex combination property, the restricted region where unstable characteristic roots of the system locate could be estimated accurately.

Further results on stability of [formula omitted

Our aim in this communication is to improve the existing results concerning the stability of xdot(t) = Ax(t) + Bx(t−τ). It is shown that the unstable characteristic roots could be more accurately located in a bounded and restrictive region of the complex plane. An iterative scheme is presented for numerical implementation of the results. Numerical computations are presented for illustration and comparison with previous results.

Further results on stability of [formula omitted

In this paper, it is shown that the restricted region where unstable characteristic roots of time-delay systems X ̇ (t) = AX(t) + BX(t − τ) exist could be more accurately estimated, and then by applying this key observation an improved delay-dependent stability criterion is proposed. A simple example is worked out to illustrate the result.

Self-Oscillation in a Control System with Dead Time and Saturation

In this paper, the author analytically investigates a second order control system with dead time and saturation and he elucidates selective occurrence of self-oscillation in the system. The conclusions obtained are as follows: The system has several pairs of unstable characteristic roots in some intervals of dead time. Among a number of self-excited modes of oscillations which correspond to unstable roots, only one mode of oscillation develops and settles down to a steady state periodic oscillation under the effect of saturation (the oscillation is referred to as self-oscillation). Although occurrence of two or more modes of self-oscillations may be possible in some intervals of dead time, these modes of oscillations can never occur simultaneously, but only one mode of them can occur depending upon the initial condition. The condition of occurrence of self-oscillation is analytically expressed and also amplitudes and frequencies of self-oscillations are theoretically obtained. The theoretical results are in good agreement with the experimental results by analog computer.