Alternative Stability Conditions Research Articles

Stability of fractional‐order switched non‐linear systems

This study addresses the stability issue of a class of fractional-order switched non-linear systems. A general stability condition is established based on which two new concepts named Mittag-Leffler increment and average Mittag-Leffler increment are proposed and two alternative stability conditions are obtained. These conditions fully reveal the trade-off between each subsystem's dynamics, and the value of the fractional order. The new results are further applied to a class of fractional-order multi-agent systems with switching cooperative control laws.

Stability of Spreading Processes over Time-Varying Large-Scale Networks

In this paper, we analyze the dynamics of spreading processes taking place over time-varying networks. A common approach to model time-varying networks is via Markovian random graph processes. This modeling approach presents the following limitation: Markovian random graphs can only replicate switching patterns with exponential inter-switching times, while in real applications these times are usually far from exponential. In this paper, we introduce a flexible and tractable extended family of processes able to replicate, with arbitrary accuracy, any distribution of inter-switching times. We then study the stability of spreading processes in this extended family. We first show that a direct analysis based on It\^o's formula provides stability conditions in terms of the eigenvalues of a matrix whose size grows exponentially with the number of edges. To overcome this limitation, we derive alternative stability conditions involving the eigenvalues of a matrix whose size grows linearly with the number of nodes. Based on our results, we also show that heuristics based on aggregated static networks approximate the epidemic threshold more accurately as the number of nodes grows, or the temporal volatility of the random graph process is reduced. Finally, we illustrate our findings via numerical simulations.

Efficient Criteria for Stability of Large-Scale Networked Control Systems

Abstract: In this paper, we analyze the stochastic stability of a class of large-scale networked control systems. We specifically consider spatially connected linear time-invariant systems whose communication with adjacent subsystems is subject to data losses. We first show that a direct application of the stability theory of Markov jump linear systems provides stability conditions in terms of the eigenvalues of a matrix whose size grows exponentially with the number of subsystems. To overcome this limitation, using the spectrum theory of random matrices, we then derive an alternative stability condition in terms of a matrix whose size grows linearly with the number of subsystems in the network. We illustrate our results with numerical simulations.

Alternative Stability Conditions for Switched Discrete Time Linear Systems

Alternative necessary and sufficient conditions for absolute exponential stability are presented, for switched discrete–time linear systems. To reach these results, we exploit concepts from set theory and in specific the forward reachability set mappings. The developed theorem can be utilized to construct iterative numerical procedures for verifying stability. Two examples illustrate the effectiveness of the proposed methodology.

Stability Analysis of Time-Delay Systems With Incommensurate Delays Using Positive Polynomials

We present a new stability condition for linear time-delay systems (TDS) with multiple incommensurate delays based on the Rekasius substitution and positive polynomials. The condition is checked by testing the positivity of multivariate polynomials in certain domains. For this purpose, we propose two alternative algorithms based on linear programming and sum of squares methods, respectively. The efficiency and accuracy of the algorithms is compared in an example to alternative stability conditions taken from the literature.

Transient and steady state of mass-conserved reaction-diffusion systems

Reaction-diffusion systems with mass conservation are studied. In such systems, abrupt decays of stripes follow quasistationary states in sequence generally. We give a stability condition of steady state which the system reaches after long transient time. It is also shown that there exist systems in which a single-stripe pattern is solely steady state for an arbitrary size of the systems. The applicability to cell biology is discussed.

Gain and phase margin of optimal memoryless regulator of systems with time-delay

A class of linear quadratic regulators of the system with time-delay in the states is constructed by a memoryless feedback, and its robustness is evaluated. First, a static non-linear perturbation is inserted in the input channel, and a robust stability condition is derived. As its special case, it is shown that the regulator has infinite gain margin and at least 50% gain reduction tolerance. Next, a dynamic linear perturbation is inserted in the same point, and an alternative stability condition is derived. It assures that the regulator has at least 60o phase margin. These results coincide with those of the finite dimensional LQ regulator.

A comment on "Decentralized stabilization of large scale interconnected systems with delays"

Hu claims new results in the above mentioned paper (ibid. vol.39, no.1, p.180-2 (1994)) regarding decentralized stabilization of large scale interconnected systems with delays. This note shows that the results presented in Theorems 1 and 2 and Corollaries 1 and 2 apply only to a very restrictive class of systems for which the number of inputs and outputs is equal to the number of states. An alternative stability condition is provided, which improves the existing one. >