Lyapunov functions provide a tool to analyze the stability of nonlinear systems without extensively solving the dynamics. Recent advances in sum-of-squares methods have enabled the algorithmic computation of Lyapunov functions for polynomial systems. However, for general large-scale nonlinear networks it is yet very difficult, and often impossible, both computationally and analytically, to find Lyapunov functions. In such cases, a system decomposition coupled to a vector Lyapunov functions approach provides a feasible alternative by analyzing the stability of the nonlinear network through a reduced-order comparison system. However, finding such a comparison system is not trivial and often, for a nonlinear network, there does not exist a single comparison system. In this work, we propose a multiple comparison systems approach for the algorithmic stability analysis of nonlinear systems. Using sum-of-squares methods we design a scalable and distributed algorithm which enables the computation of comparison systems using only communications between the neighboring subsystems. We demonstrate the algorithm by applying it to an arbitrarily generated network of interacting Van der Pol oscillators.
This contribution presents two exponential stability criteria for linear systems with multiple pointwise and distributed delays. These results (necessary and sufficient conditions) are given in terms of the delay Lyapunov matrix and the fundamental matrix of the system. An important property is that the stability test requires a finite number of mathematical operations.
In this paper, we discussed an infinitely distributed delayed viral infection model with nonlinear immune response and general incidence rate. We proved the existence and uniqueness of the equilibria. By using the Lyapunov functional and LaSalle invariance principle, we obtained the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. Numerical simulations are given to verify the analytical results.
Absolute exponential stability problem of delay time-varying systems (DTVS) with sector-bounded nonlinearity is presented in this paper. By using the comparison principle and properties of positive systems we derive several novel criteria of absolute exponential stability, for both continuous-time and discrete-time nonlinear DTVS. When applied to the time-invariant case, the obtained stability criteria are shown to cover and extend some previously known results, including, in particular, the result due to S.K. Persidskii in Ukrainian Mathematical Journal, vol. 57(2005). The theoretical results…
Input-to-state stability (ISS) allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in the stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. In this habilitation thesis, we provide a broad picture of infinite-dimensional input-to-state stability theory.
Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically computed for a large class of systems in an efficien…
This paper investigates the theory of resilience and stability in multi-microgrid networks. We derive new sufficient conditions to guarantee small-signal stability of multi-microgrids in both lossless and lossy networks. The new stability certificate for lossy networks only requires local information, thus leads to a fully distributed control scheme. Moreover, we study the impact of network topology, interface parameters (virtual inertia and damping), and local measurements (voltage magnitude and reactive power) on the stability of the system. The proposed stability certificate suggests the ex…
The traditional Wilson-Cowan model of excitatory and inhibitory mean field interactions in neuronal populations considers a weak Gamma distribution of time delays when processing inputs, and is obtained via a time-coarse graining technique that averages the population response. Previous analyses of the stability of the Wilson-Cowan model focused on more simplified cases, where the delays were either not present, constant or were of a specific type. Since these simplifications may significantly alter the behavior of the model, we focus on understanding the behavior of the system before time-cou…
We study the global dynamics of a reaction-diffusion SEIR infection model with distributed delay and nonlinear incidence rate. The well-posedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.
Inspired by Kerr's work on topological dynamics, we define tracial $\mathcal{Z}$-stability for sub-$C^*$-algebras. We prove that for a countable discrete amenable group $G$ acting freely and minimally on a compact metrizable space $X$, tracial $\mathcal{Z}$-stability for the sub-$C^*$-algebra $(C(X)\subseteq C(X)\rtimes G)$ implies that the action has dynamical comparison. Consequently, tracial $\mathcal{Z}$-stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.
Distributed Coordinated Control of Large-Scale Nonlinear Networks
利用向量李雅普诺夫函数方法，提出了一种分布式协调方法，用于大规模非线性动力系统的稳定性分析和控制设计。该方法将大系统分解为相互作用的子系统网络，通过比较系统分析系统的稳定性。然而，找到这样的比较系统并不容易。在这项工作中，我们提出了一个基于平方和的完全分散的方法来计算比较系统的网络的非线性系统。在比较系统的基础上，我们引入了一种分布式最优控制策略，在该策略中，各个子系统（智能体）与它们的近邻协调设计局部控制策略，使整个系统在初始扰动下指数稳定，并给出了相互作用的Van der Pol系统网络的控制算法.
Computation of Linear Comparison Equations for Stability Analysis of Interconnected Systems
平方和（SOS）方法已被证明是非常有用的计算多项式李雅普诺夫函数的系统，合理的小尺寸。然而，对于大规模的系统，有必要使用一个可扩展的替代使用向量李雅普诺夫函数。早期的工作表明，在一定条件下，关联系统的稳定性可以通过适当的比较方程来研究。然而，找到这样的比较方程可能是不平凡的。在这项工作中，我们提出了一个SOS为基础的系统程序直接计算比较方程的关联系统多项式动力学。通过一个相互作用的Van der Pol系统的例子，我们说明了这是如何促进稳定性分析的可扩展性和并行方法。
The death spiral hypothesis in electric utility represents a positive feedback phenomenon in which a regulated utility is driven to financial instability by rising prices and declining demand. We establish conditions for the existence of death spiral and conditions of stable adoption of distributed energy resources. We show in particular that linear tariffs always induce death spiral when the fixed operating cost of the utility rises beyond a certain threshold. For two-part tariffs with connection and volumetric charges, the Ramsey pricing that optimizes myopically social welfare subject to th…
Assessing small-signal stability of power systems composed of thousands of interacting generators is a computationally challenging task. To reduce the computational burden, this paper introduces a novel condition to assess and certify small-signal stability. Using this certificate, we can see the impact of network topology and system parameters (generators' damping and inertia) on the eigenvalues of the system. The proposed certificate is derived from rigorous analysis of the classical structure-preserving swing equation model and has a physically insightful interpretation related to the gener…
In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form the…
This paper aims at addressing distributed averaging problems for signed networks in the presence of general directed topologies that are represented by signed digraphs. A new class of improved Laplacian potential functions is proposed by presenting two notions of any signed digraph: induced unsigned digraph and mirror (undirected) signed graph, based on which two distributed averaging protocols are designed using the nearest neighbor rules. It is shown that with any of the designed protocols, signed-average consensus (respectively, state stability) can be achieved if and only if the associated…
This paper proposes methods for identification of large-scale networked systems with guarantees that the resulting model will be contracting -- a strong form of nonlinear stability -- and/or monotone, i.e. order relations between states are preserved. The main challenges that we address are: simultaneously searching for model parameters and a certificate of stability, and scalability to networks with hundreds or thousands of nodes. We propose a model set that admits convex constraints for stability and monotonicity, and has a separable structure that allows distributed identification via the a…
In this work, we have developed a Coxian distributed SEIR model in incorporating an empirical incubation period. We show that the global dynamics are completely determined by a basic reproduction number. An application of the Coxian distributed SEIR model using data of an empirical incubation period is explored. The model may be useful for resolving causing the realistic intrinsic parts in classical epidemic models since Coxian distribution approximately converges to any distribution.
