Spectral Delay Space Research Articles

An Algebraic Approach for a Stability Analysis Methodology for Multiple Time-delay Systems

This paper presents an improvement on Cluster Treatment of Characteristic Roots (CTCR), which is a well-known methodology for delay-dependent stability analysis of multiple time-delay systems (MTDS). We propose an algebraic approach to extract the stability switching hypersurfaces in spectral delay space, instead of a numerical procedure in CTCR with Extended Kronecker Sum (EKS) operation. The proposed algebraic approach is based on an efficient zero location test, and the deployment of this test to an auxiliary characteristic polynomial whose unique properties have recently been revealed. The achieved improvement is demonstrated by applying the new CTCR procedure to a system with three delays.

Investigation of Local Stability Transitions in the Spectral Delay Space and Delay Space

The stability boundaries of LTI time-delayed systems with respect to the delays are studied in two different domains: (i) delay space (DS) and (ii) spectral delay space (SDS), which contains pointwise frequency information as well as the delay. SDS is the preferred domain due to its advantageous boundedness properties and simple construct of stability transition boundaries. These transitions at the mentioned boundaries, however, present some conceptual challenges in SDS. This transition property enables us to extract the corresponding local stability variation properties in the DS, while it does not have any implication in the preferred SDS. The novel aspect of the investigation is to introduce a comparison mechanism between these two domains, DS and SDS, from the stability transition perspective. Interestingly, we are able to prove their equivalency, which provides complementary insight to the parametric stability variations.

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.

A test platform for cognitive delays: target tracking problem with multiple time-delayed feedback control

This paper studies a baseline 2-dimensional (2-D) target tracking problem from a new perspective. The dynamics entail two rationally independent time delays in an otherwise traditional feedback controller. A pursuer tries to track a moving target. When there are no time delays, capture is enabled using any stable control logic (for instance, LQR). If there are different delays in the sensing of position and velocity of the target, however, the performance of the pursuit changes dramatically. We explore the stability of the dynamics for a broad range of delay compositions. An exhaustive and exact determination of those stable delay combinations is achieved using the recent cluster treatment of characteristic roots (CTCR) paradigm. It declares the complete stability outlook in the domain of the two delays. This method starts with the determination of all the potential stability switching curves in the domain of the delays. For this, a recent concept called the “spectral delay space (SDS)” is used. As a paradoxical outcome of this work, we show that the stability of the dynamics can be recovered by increasing the delays. The paper is accompanied by an interactive MATLAB-based game which is also available online.

A consensus protocol under directed communications with two time delays and delay scheduling

This paper studies a consensus protocol over a group of agents driven by second-order dynamics. The communication among members of the group is assumed to be directed and affected by two rationally independent time delays, one in the position and the other in the velocity information channels. These delays are unknown but considered to be constant and uniform throughout the system. The stability of the consensus protocol is studied using a simplifying factorisation procedure and deploying the cluster treatment of characteristic roots (CTCR) paradigm. This effort results in a unique depiction of the exact stability boundaries in the domain of the delays. The CTCR requires the knowledge of the potential stability switching loci exhaustively within this domain. The creation of these loci is an important contribution of this work. It is done in a new surrogate coordinate system, called the spectral delay space. The relative stability of the system, that is, the speed to reach consensus is also investigated for this class of systems. Based on the outcome of this effort, a paradoxical control design concept is introduced. It is called the delay scheduling, which is another key contribution of this paper. It reveals that the performance of the system may be improved by increasing the delays. The amount of increase, however, is only revealed by the CTCR. Example case studies are presented to verify the underlying analytical derivations.

Exact stability analysis of second-order leaderless and leader–follower consensus protocols with rationally-independent multiple time delays

An investigation of double-integrator agents with directed asymmetric consensus protocols and multiple rationally independent time delays is presented in this paper from two novel perspectives. First, we complement the group consensus literature on crucial stability analysis, using a recent technique called the Cluster Treatment of Characteristic Roots (CTCR) for the first time on this class of time-delayed systems. The CTCR paradigm is pursued after a block-diagonalization (mode-decoupling) transformation on the system. This treatment produces some unique stability tables for the dynamics in the space of the delays which are non-conservative and exhaustive. Second, a novel concept of spectral delay space is presented, as an overture to the CTCR for the determination of the complete set of stability-crossing (switching) hypersurfaces in the delay space. Examples are provided to display the strengths and efficiency of this new stability analysis mechanism.

Exact Stability Analysis of a Second-Order Leaderless Consensus Protocol with Multiple Communication and Input Delays

An earlier investigation on a leaderless consensus protocol for double integrators with multiple time delays (Meng et al., 2011) is revisited in this paper from two completely different and novel perspectives. First, the crucial stability analysis of time delayed system is replaced with a recent technique called the Cluster Treatment of Characteristic Roots (CTCR). CTCR paradigm is pursued after a block-diagonalization (mode decoupling) transformation on the system. This treatment produces the unique stability tables for the dynamics in the space of the delays. Furthermore they are exact and exhaustive. Secondly, the novel concept of Spectral Delay Space is presented, as an overture to the CTCR for the determination of the needed potential stability crossing (switching) hypersurfaces in the delay space. Examples cases are provided to display the strengths, efficiency and explicitness of this new stability analysis mechanism.

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 Analysis of LTI Systems With Three Independent Delays—A Computationally Efficient Procedure

A practical numerical procedure is introduced for determining the stability robustness map of a general class of higher order linear time invariant systems with three independent delays, against uncertainties in the delays. The procedure is based on an efficient and exhaustive frequency-sweeping technique within a single loop. This operation results in determination of the complete description of the kernel and the offspring hypersurfaces, which constitute exhaustively the potential stability switching loci in the space of the delays. The new numerical procedure corresponds to the first step in the overarching framework, called the cluster treatment of characteristic roots. The results of this treatment can also be represented in another domain (called the spectral delay space) within a finite dimensional cube called the building block, which is much simpler to view and analyze. The paper also offers several case studies to demonstrate the practicality of the new numerical methodology.