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In this paper, we introduce an angle notion, called the singular angle, for stable nonlinear systems from an input-output perspective. The proposed system singular angle, based on the angle between $mathcal{L}_2$-signals, describes an upper bound for the rotating effect from the system input to output signals. It is, thus, different from the recently appeared nonlinear system phase which adopts the complexification of real-valued signals using the Hilbert transform. It can quantify the passivity and serve as an angular counterpart to the system $mathcal{L}_2$-gain. It also provides an alternative to the nonlinear system phase. A nonlinear small angle theorem, which involves a comparison of the loop system angle with $pi$, is established for feedback stability analysis. When dealing with multi-input multi-output linear time-invariant (LTI) systems, we further come up with the frequency-wise and $mathcal{H}_infty$ singular angle notions based on the matrix singular angle, and develop corresponding LTI small angle theorems.
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Discrete abstractions have become a standard approach to assist control synthesis under complex specifications. Most techniques for the construction of discrete abstractions are based on sampling of both the state and time spaces, which may not be able to guarantee safety for continuous-time systems. In this work, we aim at addressing this problem by considering only state-space abstraction. Firstly, we connect the continuous-time concrete system with its discrete (state-space) abstraction with a control interface. Then, a novel stability notion called controlled globally asymptotic/practical stability with respect to a set is proposed. It is shown that every system, under the condition that there exists an admissible control interface such that the augmented system (composed of the concrete system and its abstraction) can be made controlled globally practically stable with respect to the given set, is approximately simulated by its discrete abstraction. The effectiveness of the proposed results is illustrated by a simulation example.
In this paper we propose a novel method to establish stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) evolving according to nonlinear heterogeneous local interaction rules which is not based on Lyapunov function arguments. In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by sub-homogeneous and order-preserving nonlinear maps. This paper directly generalizes results for sub-homogeneous and order-preserving linear maps which are shown to be the counterpart to stochastic matrices thanks to nonlinear Perron-Frobenius theory. We provide sufficient conditions on the structure of local interaction rules among agents to establish convergence to a fixed point and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the method are provided to corroborate the theoretical analysis.
Recently, there have been efforts towards understanding the sampling behaviour of event-triggered control (ETC), for obtaining metrics on its sampling performance and predicting its sampling patterns. Finite-state abstractions, capturing the sampling behaviour of ETC systems, have proven promising in this respect. So far, such abstractions have been constructed for non-stochastic systems. Here, inspired by this framework, we abstract the sampling behaviour of stochastic narrow-sense linear periodic ETC (PETC) systems via Interval Markov Chains (IMCs). Particularly, we define functions over sequences of state-measurements and interevent times that can be expressed as discounted cumulative sums of rewards, and compute bounds on their expected values by constructing appropriate IMCs and equipping them with suitable rewards. Finally, we argue that our results are extendable to more general forms of functions, thus providing a generic framework to define and study various ETC sampling indicators.
We review selected results related to robustness of networked systems in finite and asymptotically large size regimes, under static and dynamical settings. In the static setting, within the framework of flow over finite networks, we discuss the effect of physical constraints on robustness to loss in link capacities. In the dynamical setting, we review several settings in which small gain type analysis provides tight robustness guarantees for linear dynamics over finite networks towards worst-case and stochastic disturbances. We also discuss network flow dynamic settings where nonlinear techniques facilitate in understanding the effect on robustness of constraints on capacity and information, substituting information with control action, and cascading failure. We also contrast the latter with a representative contagion model. For asymptotically large networks, we discuss the role of network properties in connecting microscopic shocks to emergent macroscopic fluctuations under linear dynamics as well as for economic networks at equilibrium. Through the review of these results, the paper aims to achieve two objectives. First, to highlight selected settings in which the role of interconnectivity structure of a network on its robustness is well-understood. Second, to highlight a few additional settings in which existing system theoretic tools give tight robustness guarantees, and which are also appropriate avenues for future network-theoretic investigations.
Kalman filtering has been traditionally applied in three application areas of estimation, state estimation, parameter estimation (a.k.a. model updating), and dual estimation. However, Kalman filter is often not sufficient when experimenting with highly uncertain nonlinear dynamic systems. In this study, a nonlinear estimator is developed by adopting a particle filter algorithm that takes advantage of measured signals. This approach is shown to significantly improve the ability to estimate states. To illustrate this approach, a model for a nonlinear device coupled with a hydraulic actuator plays the role of an actual plant and a nonlinear algebraic function is considered as an approximation of the nonlinear device, thus generating non-parametric and parametric uncertainties. Then we use displacement and force signals to improve the distribution of the states by resampling the set of particles. Finally, all of the states are estimated from these posterior density functions. A set of simulations considering three different noise levels demonstrates that the performance of the particle filter approach is superior to the Kalman filter, yielding substantially better performance when estimating nonlinear physical systems in the presence of modeling uncertainties.
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