Complete stability robustness of third-order LTI multiple time-delay systems

A unique procedure is presented in this paper, for a complete stability robustness of the third-order LTI multiple time-delay systems (LTI-MTDS). The uniqueness of the treatment is simply due to the fact that there is no comparable methodology, presently, in the literature. The end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays. The backbone of the method is a novel framework called “the cluster treatment of characteristic roots, (CTCR)”. CTCR is constructed over two fundamental propositions. The first proposition claims the existence of a bounded number of so-called “kernel curves”, where the only imaginary characteristic roots occur. The second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots. For simplicity of conveyance and without loss of generality, the number of time delays is taken as two in this document. The new methodology is expandable to higher-order dynamics with more time delays than two, as the authors intend to demonstrate in future publications.     

Finding the exact delay bound for consensus of linear multi-agent systems

This paper focuses on consensus problems for high-order, linear multi-agent systems. Undirected communication topologies along with a fixed and uniform communication time delay are taken into account. This class of problems has been widely studied in the literature, but there are still gaps concerning the exact stability bounds in the domain of the delays. The novelty of this paper lies in the determination of an exact and explicit delay bound for consensus. This is done in a very efficient manner by using the cluster treatment of characteristic roots (CTCR) paradigm. Before the stability analysis, a state transformation is performed to decouple the system and simplify the problem. CTCR is then deployed to the individual subsystems to obtain the stability margin in the domain of the delays without the conservatism introduced by other approaches more frequently found in the literature. Simulation results are presented to support the analytical claims.     

Consensus analysis with large and multiple communication delays using spectral delay space concept

In this study, we consider the consensus problem for a group of second-order agents interacting under a fixed, undirected communication topology. Communication lines are affected by two rationally independent delays. The first delay is assumed to be in the position information channels, whereas the second delay is in the velocity information exchange. The delays are assumed to be large and uniform throughout the entire network. The stability analysis of such systems becomes quickly intractable as the number of agents increases and the delays enlarge. To resolve this dilemma, we first reduce the complexity of the problem dramatically, by decomposing the characteristic equation of the system into a set of second-order factors. Then, we assess the stability of the resulting subsystems exactly and exhaustively in the domain of the time delays using the cluster treatment of characteristic roots (CTCR) paradigm. CTCR requires the determination of all the potential stability switching loci in the domain of the delays. For this, a surrogate domain, called the ‘spectral delay space (SDS)’, is used. The result is a computationally efficient stability analysis of the given dynamics within the domain of the delays. Illustrative cases are provided to verify the analytical conclusions. On these examples, we also study the consensus speeds through eigenvalue analysis.     

Stability of formation control using a consensus protocol under directed communications with two time delays and delay scheduling

We consider a linear algorithm to achieve formation control in a group of agents which are driven by second-order dynamics and affected by two rationally independent delays. One of the delays is in the position and the other in the velocity information channels. These delays are taken as constant and uniform throughout the system. The communication topology is assumed to be directed and fixed. The formation is attained by adding a supplementary control term to the stabilising consensus protocol. In preparation for the formation control logic, we first study the stability of the consensus, using the recent cluster treatment of characteristic roots (CTCR) paradigm. This effort results in a unique depiction of the non-conservative stability boundaries in the domain of the delays. However, CTCR requires the knowledge of the potential stability switching loci exhaustively within this domain. The creation of these loci is done in a new surrogate coordinate system, called the ‘spectral delay space (SDS)’. The relative stability is also investigated, which has to do with the speed of reaching consensus. This step leads to a paradoxical control design concept, called the ‘delay scheduling’, which highlights the fact that the group behaviour may be enhanced by increasing the delays. These steps lead to a control strategy to establish a desired group formation that guarantees spacing among the agents. Example case studies are presented to validate the underlying analytical derivations.     

Exhaustive stability analysis in a consensus system with time delay and irregular topologies

A consensus problem and its stability are studied for a group of agents with second-order dynamics and communication delays. The communication topologies are taken as irregular but always connected and undirected. The delays are assumed to be quasi-static and the same for all the interagent channels. A decentralised, PD-like control structure is proposed to create a consensus in the position and velocity of the agents. We present an interesting factorisation feature for the characteristic equation of the system which simplifies the stability analysis considerably from a prohibitively large dimensional problem to a manageable small scale. It facilitates a rare stability picture in the space of the control parameters and the delay, utilising a paradigm named cluster treatment of characteristic roots (CTCR). The influence of the individual factors on the absolute and relative stability of the system is studied. This leads to the introduction of two novel concepts: the most exigent eigenvalue, which refers to the one that defines the delay stability margin of the system, and the most critical eigenvalue, which is the one that dictates the consensus speed of the system. It is observed that the most exigent eigenvalue is not always the most critical, and this feature may be used as a design tool for the control logic. Case studies and simulations results are presented to verify these concepts.     

Stability Robustness Analysis of Multiple Time- Delayed Systems Using “Building Block” Concept

An intriguing perspective is presented in studying the stability robustness of systems with multiple independent and uncertain delays. It is based on a holographic mapping, which is implemented over the domain of the delays. This mapping considerably alleviates the problem, which is otherwise known to be notoriously complex. It creates a dramatic reduction in the dimension of the problem from infinity to manageably small number. Ultimately the process is reduced to studying the problem within a finite dimensional cube with edges of length 2pi in the new domain, what we call the building block. In essence, the mapping collapses the entire set of potential stability switching points onto a small (upperbounded) number of building hypersurfaces. We further demonstrate that these building hypersurfaces can be implicitly defined and they are completely isolated within the above mentioned cube. It is also shown that the exhaustive detection of these building hypersurfaces is necessary and sufficient in order to arrive at the complete stability robustness picture we seek. As a consequence, this concept yields a very practical and efficient procedure for the stability assessment of such systems. This novel perspective serves very well for the preparatory steps of the authors' earlier contribution in the area, cluster treatment of characteristic roots (CTCR). We elaborate on this combination, which forms the main contribution of the paper. Several example case studies are also provided     

Control of time delay processes with uncertain delays: Time delay stability margins

In this paper, the stability of time delay processes that have uncertain delays is considered, and the maximum allowable perturbation which may occur in the time delays so as to maintain stability are determined. In particular, the characteristic equations of time delay systems are quasipolynomials, whose roots determine the stability of such systems, and the root-locus of these equations in specified desired regions is investigated. A numerical algorithm is presented for the calculation of the time delay stability margins in the space of time delays for such systems, and the size of the stability hyperspheres in this space is computed. To illustrate the procedure, the algorithm is applied to process control systems with uncertain delays and the allowable perturbations in the time delays of these systems are then computed.     

带有通信时滞的二阶多智能体系统一致问题

针对具有双向等时延的二阶无向通信拓扑系统,采用带有通信时滞的线性一致控制率协议,分析了使系统稳定的条件。由于系统的阶次较高,直接对其特征方程进行分析是比较困难的,提出了一种新的分析方法,把系统的特征方程分解为多个子系统的乘积,然后利用CTCR方法,求得每个子系统对应的时滞最大值,比较后得出使系统达到一致稳定的最大时滞,作出了控制率边界曲线图并标出了稳定区域。结果表明,在有向生成树的情况下,当时滞小于决策值时,系统能达到稳定。最后,数值仿真验证了所得结果的有效性。     

Stability independent of delay using rational functions

This paper is concerned with the problem of assessing the stability of linear systems with a single time-delay. Stability analysis of linear systems with time-delays is complicated by the need to locate the roots of a transcendental characteristic equation. In this paper we show that a linear system with a single time-delay is stable independent of delay if and only if a certain rational function parameterized by an integer k and a positive real number T has only stable roots for any finite T≥0 and any k≥2. We then show how this stability result can be further simplified by analyzing the roots of an associated polynomial parameterized by a real number δ in the open interval (0,1). The paper is closed by showing counterexamples where stability of the roots of the rational function when k=1 is not sufficient for stability of the associated linear system with time-delay. We also introduce a variation of an existing frequency-sweeping necessary and sufficient condition for stability independent of delay which resembles the form of a generalized Nyquist criterion. The results are illustrated by numerical examples.     

依赖延迟线性时滞系统的稳定性判据

由于独立延迟线性时滞（Linear time-delay with independent delays,LTD-ID）系统的稳定条件对系统参数有严格的限制,只有极少数依赖延迟线性时滞（LTD with dependent delays,LTD-DD）系统可满足该稳定条件. LTD-ID 系统的特征多项式属拟多项式,其根为多重延迟的函数,这使得LTD-ID系统的稳定性检验非常困难. 为解决该问题,基于二维域混合多项式,本文提出LTD-DD系统的若干稳定性判据. 应用例表明所提出的稳定性判据是简单的和有效的,所提出的定理4可解决现有LMI稳定性判据的保守性问题.     

Stability and L∞‐gain analysis for a class of nonlinear positive systems with mixed delays

This paper addresses the asymptotic stability and L_∞‐gain analysis problem for a class of nonlinear positive systems with both unbounded discrete delays and distributed delays. With the assumption that the nonlinear function is strictly increasing, we first give a characterization on the positivity of the nonlinear system. Then, with some mild assumptions on the delays, a necessary and sufficient condition to ensure the asymptotic stability is presented. Moreover, an explicit expression of the L_∞‐gain of such nonlinear positive systems is given in terms of the system matrices. Finally, a numerical example is given to illustrate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Further stability results of a delay-differential systems for digital networks

A criterion is given which is sufficient for the characteristic roots to remain within the semi-plane Rc (λ)<0. It is not necessary to find the roots of transcendental equation but, the traditional Gorshgorin criterion for each t≥t₀, together with another condition are sufficient to determine the stability of tho IJ.G.M. systems. Dopplor effects due to variable time delays in Sandborg's non-linear model is being considered.     

Generalization of cluster treatment of characteristic roots for robust stability of multiple time‐delayed systems

A new perspective is presented for studying the stability robustness of nth order systems with p rationally independent delays. It deploys a holographic mapping procedure over the delay space into a new coordinate system in order to achieve the objective. This mapping collapses the entire set of potential stability switching points on a manageably small number of hypersurfaces, which are explicitly defined in the new domain. This property considerably alleviates the problem, which is otherwise infinite dimensional, and therefore notoriously complex to handle. We further declare some unrecognized features of these switching hypersurfaces, that they are (a) encapsulated within a higher‐dimensional cube with edges of length 2π, which we name the ‘building block’, and (b) the ‘offspring’ of this building block, which represent the secondary stability switchings, appear within the adjacent and identical building blocks (cubes) stacked up next to each other. The final outlook is an exclusive representation of stability for this general class of systems at any arbitrary point in the delay space. Two example case studies are also provided, which are not possible to analyze using any other methodology known to the authors. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability and performance analysis of linear positive systems with delays using input–output methods

It is known that input–output approaches based on scaled small-gain theorems with constant D-scalings and integral linear constraints are non-conservative for the analysis of some classes of linear positive systems interconnected with uncertain linear operators. This dramatically contrasts with the case of general linear systems with delays where input–output approaches provide, in general, sufficient conditions only. Using these results, we provide simple alternative proofs for many of the existing results on the stability of linear positive systems with discrete/distributed/neutral time-invariant/-varying delays and linear difference equations. In particular, we give a simple proof for the characterisation of diagonal Riccati stability for systems with discrete-delays and generalise this equation to other types of delay systems. The fact that all those results can be reproved in a very simple way demonstrates the importance and the efficiency of the input–output framework for the analysis of linear positive systems. The approach is also used to derive performance results evaluated in terms of the L ₁-, L ₂- and L _∞-gains. It is also flexible enough to be used for design purposes.     

Robust H∞ controller design for a class of linear quantum systems with time delay

Time delay is frequently encountered in practical quantum feedback control systems with long transmission lines and measurement process. This paper is concerned with measurement‐based feedback H^∞ control for quantum systems with time delays appearing in the feedback loops. A physical model is presented for the quantum time‐delay system described by complex quantum stochastic differential equations. Quantum versions of some fundamental properties, such as dissipativity and stability, are discussed for this model. A numerical procedure is proposed for H^∞ controller synthesis, which can deal with a non‐convex optimization problem arising in the design processes. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust H∞ control for networked systems with transmission delays and successive packet dropouts under stochastic sampling

This paper is concerned with the H_∞ performance analysis for networked control systems with transmission delays and successive packet dropouts under stochastic sampling. The parameter uncertainties are time‐varying norm‐bounded and appear in both the state and input matrices. If packet loss is considered the same as time delay, when models the networked control systems with successive packet dropouts and delays as ordinary linear system with input‐delay approach, due to sampling period is stochastic, then the delay caused by packet losses is a stochastic variable, which leads to difficulties in the stability analysis of the considered system. However, if we can transform the system with stochastic delay into a continuous system with stochastic parameter, we can solve the problem. In this paper, by assuming that the network packet loss rate and employing the information of probabilistic distribution of the time delays, the stochastic sampling system is transformed into a continuous‐time model with stochastic variable, which satisfies a Bernoulli distribution. By linear matrix inequality approach, sufficient conditions are obtained, which guarantee the robust mean‐square exponential stability of the system with an H_∞ performance. What's more, an H_∞ controller design procedure is then proposed, and a less conservative result is obtained by taking the probability into consideration. Finally, a numerical simulation example is employed to show the effectiveness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.     

New results on stability and L∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays

This article addresses the problems of stability and L_∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays for the first time. First, we consider the stability problem of a class of positive linear differential‐algebraic equations with unbounded time‐varying delays. A new method, which is based on the upper bounding of the state vector by a decreasing function, is presented to analyze the stability of the system. Then, by investigating the monotonicity of state trajectory, the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is characterized. It is shown that the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is also independent of the delays and fully determined by the system matrices. Two numerical examples are given to illustrate the obtained results.     

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.     

Stability crossing surfaces for linear time-delay systems with three delays

In this article, the stability of linear systems with time-delays is studied. The cases where the characteristic equations of the system include three delays are investigated. Using the geometrical relations in a normalised polynomial plane, a graphical method is presented to visualise the stability domains in the space of the time-delays. In this space, the points at which the characteristic equation has a zero on the imaginary axis (the border between stability and instability regions) are identified. These points form several surfaces called the ‘stability crossing surfaces’. This work extends the results of the previous works on the ‘stability crossing curves’ defined in the two-dimensional space (plane) of delays to a higher dimension and provides new geometric interpretations for the stability crossing conditions.