Obtaining the Puiseux series of multiple imaginary (characteristic) roots (MIRs) is a fundamental issue in stability analysis time-delay systems. However, to best authors' knowledge, this has not been fully investigated up date. This note focuses on expansion MIRs linear time-invariant systems including commensurate delays. For an MIR any multiplicity, we propose algorithm for defining structure series, as well explicit computation corresponding coefficients. By using proposed method, can...

Abstract This paper introduces a novel Jacobian‐based frequency‐sweeping approach for constructing stability map on delay plane linear time‐invariant (LTI) systems with two delays. It is demonstrated that by imposing zero Jacobian condition the Rekasius‐transformed polynomial equations defining crossing curves (SCCs), set of frequencies (SCFs) can be determined exactly and exhaustively. Compared to existing frameworks use discrimination criterion identify upper bound SCFs, our proposed...

In this paper, a nearly data-based optimal control scheme is proposed for linear discrete model-free systems with delays. The can be obtained using only measured input/output data from systems, by reinforcement learning technology, which combines Q-learning value iterative algorithm. First, we construct state estimator the data. Second, quadratic functional used to approximate function at each point in space, and designed method estimator. Then, paper states method, that is, how solve inner...

The coexistence of multiple stable equilibria in recurrent neural networks is an important dynamic characteristic for associative memory and other applications. In this paper, the existence local Mittag-Leffler stability are investigated a class fractional-order with discontinuous nonmonotonic activation functions. By using Brouwer's fixed point theory, several conditions established to ensure 5 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML"...

The resultant-based frequency-sweeping approach is appealing for creating a stability map to study robustness against delay uncertainties. It relies on converting the system's transcendental characteristic function into polynomial form such that crossing frequency set and corresponding switching curves in parameter plane can be exactly exhaustively determined using only algebraic computations. In existing literature, equation conversion has been achieved by Rekasius substitution, bilinear or...

The paper proposes a novel procedure for the asymptotic expansions of root loci around multiple imaginary roots an exponential polynomial, which is necessary stability analysis LTI systems with commensurate delays. With delay given as polynomials (also called quasi-polynomial), we seek to characterise behaviours characteristic such in algebraic way and determine whether cross from one half-plane into another or only touch axis. According Weierstrass preparation theorem, quasi-polynomial...

Abstract The stabilization of first‐order delayed (FOD) unstable processes with proportional–integral–derivative (PID) controllers is considered, and all the feasible PID are determined. Different from existing results which based on D‐partition technique partitioning complex's real imaginary parts, a novel procedure enlightened by τ decomposition method proposed to characterize space controller parameters. Generally speaking, parameter mainly divided into four regions: delay independent...

This paper considers the Modified Autonomous Van der Pol–Duffing equation subjected to dynamic state feedback, which can well characterize behaviors of nonlinear dynamical systems. Both issues local stability switches and Hopf bifurcation versus time delay are investigated. Associating with τ decomposition strategy center manifold theory, stable intervals direction all determined. Specifically, computation purely imaginary roots (symmetry real axis), positive root formula for cubic...

This paper reveals novel properties of spectrum linear delay differential systems, and concerns the stability a general class time systems. The issue can be well characterized by distribution associated characteristic equation. Benefit on finite bounded are formulated. And finiteness condition boundary obtained in practical way. Based argument principle, an analytical formula for computing number unstable zeros (with positive real part) is deduced, which determined definite integral....

The central task of the stability robustness with respect to delay uncertainties lies in creating whole chart (delay map) parameter space. In this paper, we present a very simple frequency-sweeping procedure for plane class linear time-invariant (LTI) two-delay systems crossing talk. Actually, based on using Rekasius pseudo-delay substitution and discriminant quadratic polynomial characterize frequency set. exact exhaustive determination intervals involves only finding all positive real...

In this study, we investigate a class of Fast Transmission Control Protocol (FAST TCP) network with delayed state feedback mechanism in terms nonlinear dynamics, including local stability and periodic bifurcation. First, successfully addressed the system by employing τ -decomposition strategy provided an explicit expression for delay interval. Second, apply normal form theory central manifold theorem to study bifurcation solutions depth clarify direction Hopf as well solutions. addition,...

In this paper, multistability is discussed for delayed recurrent neural networks with ring structure and multi-step piecewise linear activation functions. Sufficient criteria are obtained to check the existence of multiple equilibria. A lemma proposed explore number cross-direction purely imaginary roots characteristic equation, which corresponds network model. Stability all equilibria investigated. The work improves extends existing stability results in literature. Finally, two examples...

This study presents an effective analysis procedure to give the bounded input output stability intervals of time delay for a class commensurate delayed fractional‐order systems with rational order. First, it is proposed that on multi‐valued characteristic function can be conducted equivalent consideration principal branch. The statement makes generalisation τ ‐decomposition method in integer order case possible. With convenience studying systems, frequency‐sweeping applied logically, and...

This paper considers the problems of determining complete stabilising set proportional-derivative controllers for a first-order process with time delay. First, by employing version Hermite–Biehler theorem applicable to quasi-polynomials, all parameters processes constant delay are obtained. Next, we provide an approach design robust PD controller stabilise uncertain delay, which lies inside known interval.

This paper investigates the local stability, existence and stability of Hopf bifurcation typical logistic differential equation versus time delay. The is considered within framework $\tau$ decomposition method, which involves calculation PIR determination cross direction around it. And complete exact delay stable interval given analytically. Subsequently, nonlinear dynamics bifurcating solutions are reviewed carefully by center manifold theorem normal form theory. By way, a new simple...