This paper provides a framework to characterize the gain margin, phase margin, and maximum input delay margin of a linear time-invariant multi-agent system where the interaction topology is described by a graph with a directed spanning tree. The stability analysis of the multi-agent system based on the generalized Nyquist theorem is converted to finding a minimum gain positive definite Hermitian perturbation and minimum phase unitary perturbation in the feedback path of the loop transfer function. Specifically, two constrained minimization problems are solved to calculate the gain, phase, and input delay margins of the multi-agent system. We further state necessary and sufficient conditions concerning stability of the multi-agent system independent of gain and phase perturbations, and input delay.
High penetration of photovoltaic (PV) generators can lead to voltage issues in distribution networks. Various approaches including the real power control through PV inverters have been proposed to address voltage issues. However, among different control strategies, communication delays are inevitably involved and they need to be carefully considered in the control loop. Those delays can significantly deteriorate the system performance with undesired voltage quality, and may also cause system instability. In this paper, according to the inverter based active power control strategy, a linearized…
We study the delay margin problem in the context of recent works by T. Qi, J. Zhu, and J. Chen, where a sufficient condition for the maximal delay margin is formulated in terms of an interpolation problem obtained after introducing a rational approximation. Instead we omit the approximation step and solve the same problem directly using techniques from function theory and analytic interpolation. Furthermore, we introduce a constant shift in the domain of the interpolation problem. In this way we are able to improve on their lower bound for the maximum delay margin.
We show that the open-loop transfer functions and the stability margins may be defined within the recent model-free control setting. Several convincing computer experiments are presented including one which studies the robustness with respect to delays.
Delay-dependent passivity and stability of linear port-Hamiltonian systems
ENSAM, Hassan II University of Casabalanca（卡萨布兰卡哈桑二世大学ENSAM）；University of Wuppertal（伍珀塔尔大学）；Fraunhofer IEG, Fraunhofer Research Institution for Energy Infrastructures and Geotechnologies IEG（弗劳恩霍夫能源基础设施和岩土技术研究所IEG）
Motivated by the regulator theory and adaptive controls, several notions on output stability in the framework of input-to-state stability (iss) were introduced for finite-dimensional systems. It turned out that these output stability notions are intrinsically different, reflecting different manners of how state variables may affect the transient behavior of output variables. In this work, we consider these output stability properties for delay systems. Our main objective is to illustrate how the various notions are related for delay systems and to provide the Razumikhin criteria for the output…
We study exponential stability for a kind of neural networks having time-varying delay. By extending the auxiliary function-based integral inequality, a novel integral inequality is derived by using weighted orthogonal functions of which one is discontinuous. Then, the new inequality is applied to investigate the exponential stability of time-delay neural networks via Lyapunov-Krasovskii functional (LKF) method. Numerical examples are given to verify the advantages of the proposed criterion.
In Part I of this paper, we have developed a novel $\mathcal{L}_1$ adaptive control architecture that enables fast adaptation and leads to uniformly bounded transient and asymptotic tracking for system's both signals, input and output, simultaneously. In this paper, we derive the stability margins of $\mathcal{L}_1$ adaptive control architecture, including time-delay and gain margins in the presence of time-varying bounded disturbance. Simulations verify the theoretical findings.
This paper studies the problem of stability of a parameterized delay differential equations (DDE see equation (0.1)). After discretizing the DDE (0.1), we show that the problem can be equivalently casted into a semi-definite programming (SDP) see (3.2), which can be solved efficiently through some popular algorithm, e.g., the interior point method [1].
This paper studies a class of random nonlinear systems with time-varying delay, in which the $r$-order moment ($r\geq1$) of the random disturbance is finite. Firstly, some general conditions are proposed to guarantee the existence and uniqueness of the global solution to random nonlinear time-delay systems. Secondly, some definitions and criteria on noise-to-state stability in the moment sense and in probability sense are given by Lyapunov method respectively. Finally, two regulation controllers are constructed respectively for two corresponding random nonlinear time-delay systems and the effe…
The stability of feedback systems consisting of linear time-delay plants and PID controllers has been investigated for many years by means of several methods, of which the Nyquist criterion, a generalization of the Hermite-Biehler Theorem, and the root location method are well known. The main purpose of these researches is to determine the range of controller parameters that allow stability. Explicit and complete expressions of the boundaries of these regions and computation procedures with a finite number of steps are now available only for first-order plants, provided with one time delay. In…
Unlike the situation with gain and phase margins in robust stabilization, the problem to determine an exact maximum delay margin is still an open problem, although extensive work has been done to establish upper and lower bounds. The problem is that the corresponding constraints in the Nyquist plot are frequency dependent, and encircling the point $s=-1$ has to be done at sufficiently low frequencies, as the possibility to do so closes at higher frequencies. In this paper we present a new method for determining a sharper lower bound by introducing a frequency-dependent shift. The problem of fi…
The aim of this paper is to give a presentation of the Python toolbox YALTAPy dedicated to the stability study of standard and fractional delay systems as well as its online version YALTAPy_Online. Both toolboxes are derived from YALTA whose functionalities will be recalled here. Examples will be given to show how these toolboxes may be used.
This paper deals with the exponential stabilization of a time-delay system with an average of the state as the output. A general stability theorem with a guaranteed exponential decay-rate based on a Wirtinger-based inequality is provided. Variations of this theorem for synthesis of a controller or for an observer-based control is derived. Some numerical comparisons are proposed with existing theorems of the literature and comparable results are obtained but with an extension to stabilization.
Based on the classical probability, the stability criteria for stochastic differential delay equations (SDDEs) where their coefficients are either linear or nonlinear but bounded by linear functions have been investigated intensively. Moreover, the dependent stability of the highly nonlinear hybrid stochastic differential equations is recently studied. In this paper, by using the nonlinear expectation theory, we explore the dependent stability of a class of highly nonlinear hybrid stochastic differential delay equations driven by $G$-Brownian motion ($G$-SDDEs). Firstly, we give preliminaries …
This paper considers $L_2$ and BIBO stability and stabilization issues for systems with time-varying delays which can be of retarded or neutral type. An important role is played by a nominal system with fixed delays which are close to the time-varying ones. Under stability or stabilizability conditions of this nominal system, sufficient conditions are given in order to ensure similar properties for the system with time-varying delays.
We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parametrization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.
The present paper is mainly aimed at introducing a novel notion of stability of nonlinear time-delay systems called Rational Stability. According to the Lyapunov-type, various sufficient conditions for rational stability are reached. Under delay dependent conditions, we suggest a nonlinear time-delay observer to estimate the system states, a state feedback controller and the observer-based controller rational stability is provided. Moreover, global rational stability using output feedback is given. Finally, the study presents simulation findings to show the feasibility of the suggested strateg…
This paper is devoted to the stability analysis of an n species Lotka-Volterra system with discrete and distributed delays. Stochastic perturbations to the parameters of the model are allowed. Sufficient conditions for the almost sure global asymptotic stability of the positive equilibrium are derived in terms of LMIs. The efficiency of the proposed method is illustrated by numerical examples.
Stability of linear systems with uncertain bounded time-varying delays is studied under assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the $L_2$-norm of a certain integral operator. There exists a bound on this operator in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving an operator bound which is an increasing and continuous functi…
This paper considers the robust ${\cal D}$-stability margin problem under polynomic structured real parametric uncertainty. Based on the work of De Gaston and Safonov (1988), we have developed techniques such as, a parallel frequency sweeping strategy, different domain splitting schemes, which significantly reduce the computational complexity and guarantee the convergence.
Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimonotonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thu…
