October 16, 2014
Abstract
This paper deals with the development **and** the **analysis** of asymptotic stable **and** consistent schemes in the joint quasi-neutral **and** fluid limits for the collisional Vlasov-Poisson **system**. In these limits, the classical explicit schemes suffer from time step restrictions due to the small plasma period **and** Knudsen number. To solve this problem, we propose a new scheme stable for choices of time steps independent from the small scales dynamics **and** with comparable computational cost with respect to stan- dard explicit schemes. In addition, this scheme reduces automatically to consistent discretizations of the underlying asymptotic systems. In this first work on this subject, we propose a first order in time scheme **and** we perform a relative linear **stability** **analysis** to deal with such problems. The framework we propose permits to extend this approach to high order schemes in the next future. We finally show the capability of the method in dealing with small scales through numerical experiments.

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Keywords: hybrid systems, nonlinear systems, Coulomb friction, Lyapunov methods, global asymptotic **stability**, PID control.
1. INTRODUCTION
In this paper, we present a hybrid model formulation for a proportional-integral-derivative (PID) controlled single- mass motion **system** subject to Coulomb friction. Friction is a performance-limiting factor in many high-precision positioning systems in terms of the achievable setpoint accuracy **and** settling times. Many different control strate- gies have been developed for frictional systems (e.g., model-based compensation, see Armstrong-H´ elouvry et al. (1994), impulsive control, see van de Wouw **and** Leine (2012), or sliding mode control, see Bartolini et al. (2003)). PID control, however, is still used in the vast majority of industrial motion systems, since the integrator action is able to compensate for unknown static friction. However, it has performance limitations (e.g., long settling times, see Beerens et al. (2018)). The popularity of the PID controller for mechatronics applications in industry, how- ever, motivates the development of hybrid add-ons, such as reset control strategies, to complement the classical PID controller **and** improve its baseline performance.

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All generators except g13, g19 **and** g20 are equipped with PSS using the rotor speed ω as input (a zero value for K p in Table 13 indicates the absence of PSS). ω is in per unit. Each PSS includes a washout filter **and**
two identical lead filters in cascade. The PSS phase compensation was chosen considering the maximum **and** minimum equivalent Th´evenin impedances seen by the machines of each group (units 7 **and** 18 for the round-rotor, units 4 **and** 12 for the salient-pole machines). The PSS transfer functions provide damping for oscillation frequencies from 0.2 Hz to more than 1 Hz.

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While polynomial approximation methods for the **analysis** of infinite dimension systems is not a new idea (see for in- stance the convex optimization **and** sum-of squares frame- works developed in Papachristodoulou **and** Peet (2006); Peet (2014) or Ahmadi et al. (2014)), the novelty of this approach relies on the use of efficient integral inequalities that are able to give a measure of the conservatism as- sociated to the approximation. These inequalities can be interpreted as a Bessel inequality on Hilbert spaces. In previous work, e.g. Seuret **and** Gouaisbaut (2014, 2015), the efficiency of these inequalities for the **stability** **analysis** of TDS has been shown. Indeed, one can also read in Seuret et al. (2015) a method based on a polynomial approximation of the distributed nature of the delay, using Legendre polynomials **and** their properties to construct Lyapunov-Krasovskii functionals. In the present paper, where a simple transport equation replaces the delay terms (an approach also studied in e.g. Bekiaris-Liberis **and** Krstic (2013)), an alternative use of this new method is proposed.

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More generally, this **system** can be seen either as the control of the PDE by a finite dimensional dynamic control law generated by an ODE [8] or on the contrary the robustness of a linear closed loop **system** with a control signal conveyed by a damped string equation. On the first scenario, both the ODE **and** the PDE are stable **and** the **stability** of the coupled **system** is studied. The second case corresponds to an unstable but stabilizable ODE connected to a stable PDE. To sum up, this paper focuses on the **stability** **analysis** of closed-loop coupled **system** (1) with a potentially unstable closed-loop ODE but a stable PDE. This differs significantly from the backstepping methodology of [15] which aims at designing an infinite dimensional control law ensuring the **stability** of a cascaded ODE-PDE **system** with a closed-loop stable ODE. A. Existence **and** regularity of solutions

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Abstract: The objective of this contribution is to improve recent **stability** results for a **system** coupling ordinary differential equations to a vectorial transport partial differential equation by proposing a new structure of Lyapunov functional. Following the same process of most of the investigations in literature, that are based on an a priori selection of Lyapunov functionals **and** use the usual integral inequalities (Jensen, Wirtinger, Bessel...), we will present an efficient method to estimate the exponential decay rate of this coupled **system** leading to a tractable test expressed in terms of linear matrix inequalities. These LMI conditions stem from the new design of a candidate Lyapunov functional, but also the inherent properties of the Legendre polynomials, that are used to build a projection of the infinite dimensional part of the state of the **system**. Based on these polynomials **and** using the appropriate Bessel-Legendre inequality, we will prove an exponential **stability** result **and** in the end, we will show the efficiency of our approach on academic example.

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for a given function h : [0, ∞) → R.
This suggests to search for stationary solutions of (1)–(4) as minimizers of a Lya- pounov functional using (8), (9) **and** (10). The natural energy (7) does not have enough structure to serve as a Lyapounov functional for the evolution problem. The idea is thus to combine (7) **and** (10) **and** to study the critical points of this Casimir- energy functional, under a mass constraint. It turns out that the critical points are also stationary solutions of (1)–(4). Moreover, the minimization property defines the framework that allows us to investigate the **stability** issue. We refer for instance the reader to [3, 6, 12, 18, 26] for similar reasonings, see also the overview [23]. We proceed as follows. Let h : [0, ∞) → R be a strictly convex function. We set

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∗ LAAS-CNRS, Universit´ e de Toulouse, CNRS, UPS, Toulouse,
France.
Abstract: This paper addresses the **stability** **analysis** of a **system** of ordinary differential equations coupled with a classic heat equation using a Lyapunov approach. Relying on recent developments in the area of time-delay systems, a new method to study the **stability** of such a class of coupled finite/infinite dimensional systems is presented here. It consists in a Lyapunov **analysis** of the infinite dimensional state of the **system** using an energy functional enriched by the mean value of the heat variable. The main technical step relies on the use an efficient Bessel-like integral inequality on Hilbert space leading to tractable conditions expressed in terms of linear matrix inequalities. The results are then illustrated on academic examples **and** demonstrate the potential of this new approach.

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The idea of using Simulink to develop an educational pack- age for the **analysis** of Power **System** Dynamics first arose in 1997 at ULg **and** matured progressively, in particular within the context of a collaboration with the University of Bologna [7]. On the lines of this original idea various modules simulat- ing power **system** components have been developed at ULg, as well as in the Electrical Energy Systems Lab of NTUA. These include generic models of traditional components (synchro- nous generator, AVR, governor, voltage-dependent loads, in- duction motor, SVC, etc.), as well as specific models for FACTS devices, steam **and** hydro power plants, wind parks

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3.6 Observation of chaotic dynamics
This nonlinear **system** exhibits the double-scroll chaotic attractor. It should be pointed out that, with different parameters, **system** (3) can evolve to other complex dynamics such as single-scroll chaotic attractor **and** periodic orbit (Huang **and** Zhao, 2016). In this **system**, the capacitor C is a parameter of control **and** the value of C can be changed within a certain range. When the parameter C is changed, the chaos behaviour of this **system** can effectively be controlled. In the numerical simulation, the initial values of the **system** (3) are (0, 5, 4). Using MATLAB program, the numerical simulation have been completed. This nonlinear **system** exhibits the complex **and** abundant of the chaotic dynamics behaviours, the strange attractors (Tremori et al., 2016) are shown in Figure 6. These phase portrait are obtained by solving equations (3) by means of Runge-Kutta method for step size of 0.000001. At the beginning of simulation u 1 = u 2 = 0. Now is clear that the

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E5A Ca 2+ (from left to right), with respect to the wild-type (HSE) pilus. The differences are then mapped on the structure of each **system**. The 297
mutation positions are shown with yellow sticks **and** pointed to by black arrows. (D) The distances between pairs of residues forming salt
298

Power systems are evolving towards massive penetration of renewable energy resources, including utility-scale **and** distributed photovoltaic (PV) plants, to leverage their economic **and** environmental advantages [1]. Regarding the operation of PV-rich power systems, one of the main issues is related to frequency instability which corresponds to generation-load mismatch **and** may lead to cascading failures in the form of generation trip, load shedding, or even splitting of the **system** into islanded areas [2]. Technically, a **system** with adequate frequency control ancillary services (FCAS) is more likely to regain a stable equilibrium point following frequency contingencies [2]. However, the increase of PV penetration **and** the subsequent decline in synchronous generation drives power systems into low-inertia conditions which may result in higher rate of change of frequency (RoCoF) values following disturbances, also meaning faster frequency dynamics in general. It is therefore necessary to model the **system** adequately to be able to capture such fast frequency behaviour. For short-term frequency **stability** **analysis**, electromagnetic transient (EMT) models or transient **stability** models have been mostly employed, which comprise numerous differential-algebraic equations (DAEs) to provide proper representation of power **system** components [3]. Moreover, there are practical evidence that transmission- connected PV units are required by grid-codes to participate in primary frequency response while the frequency is beyond the normal frequency operating band [4], [5]. Hence, it is also

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a transmission line modeled by hyperbolic PDEs. Since we combine both dynamics, the coupling is defined at the boundaries of the hyperbolic PDEs. It results in what we call, coupled PDE-ODE with dynamic boundary condition. We aim then at studying the **stability** of such systems un- der the assumption that one wants the **system** to operate in free-flow zone to avoid congestion. The main contribution of this work is the modeling **and** the study of input- to-state **stability** properties of a communication network when operating at some optimal equilibrium point. This paper is organized as follows. In Section 2 we in- troduce the proposed model. Section 3 contains the well- posedness **and** **stability** **analysis**, including the steady-state characterization. Section 4 provides a numerical example to illustrate the results. Finally, conclusions are given in Section 5.

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proliferation of CIGs, faster dynamics will gain more prominence when analyzing future power **system** dynamic behavior compared to the phenomena within the time scale of several milliseconds to minutes. Focusing on the time scale of the electromechanical transients enabled several simplifications in power **system** modeling **and** representation, which significantly aided the characterization **and** **analysis** of the related phenomena. A key aspect of these simplifications is the assumption that voltage **and** current waveforms are dominated by the fundamental frequency component of the **system** (50 or 60 Hz). As a consequence, the electrical network could be modeled considering steady-state voltage **and** current phasors, also known as a quasi-static phasor modeling approach. With this modelling approach, high- frequency dynamics **and** phenomena, such as the dynamics associated with the switching of power electronic converters, are only represented by either steady-state models or simplified dynamic models, meaning that fast phenomena, like switching, cannot be completely captured. Considering the CIG related time scales of operation mentioned previously, there is a need to extend the bandwidth of the phenomena to be examined **and** include faster dynamics within electromagnetic time scales when the faster dynamics is of importance **and** can affect overall **system** dynamics.

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The next objective of data mining will be to determine the main weak points of the power **system** being studied (i.e., the relative frequency of different types of behaviors: stable, voltage collapse, cascading line tripping, loss of synchronism, frequency collapse, oscillations, dynamic instabilities...). This can be achieved using automatic un- supervised learning techniques, as illustrated in Section 5. Finally, further steps will consist of assessing how the different variables used to characterize the scenarios are interrelated, **and** will automatically identify those that are useful to detect various undesirable phenomena. In particular, supervised learning **and** correlation **analysis** may be used to screen automatically a large number of variables **and** scenarios relevant for each task. Notice that at this step the engineer can try out various possibili- ties, being limited only by his imagination **and** the in- formation contained in the database. In order to find efficient ways to predict **system** failures, he may try out different ways of decomposing the overall problem into subproblems, **and** test different sets of measurements **and** detection logics. Such detection rules may be defined either manually or automatically using supervised learn- ing methods.

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More precisely, after a contingency inception **and** its clearance, SIME drives a time-domain program so as to accomplish the fol- lowing tasks: identify the critical **and** non critical machines **and** aggregate them into two groups; replace these groups by succes- sively a two-machine, then an OMIB equivalent **system**; assess transient **stability** of this OMIB, using the EAC [6]. The various steps of the method are briefly described below **and** illustrated in Figs 1 drawn for the 3-machine **system** simulated in Section 5. For more details about SIME, see [7].

II. B OND G RAPH B ASIC P RINCIPLES
The BG formalism [1-3], introduced by H. Paynter in 1961 **and** formalised by Karnopp **and** Rosenberg in 1975, is now regularly used in many companies, particularly in automobile industry (PSA, Renault, Ford, Toyota, General Motors, …). This graphical method illustrates the energetic transfers in the **system**. The orientation of a half arrow indicates the power flow as expressed in Fig. 1. The primary characteristic of the BG is relative to its unified aspect through analogies summarised in Table 1. It obviously shows that this method can be applied to all physical domains by using generalised variables for :

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Figure 2(a) obviously shows the stable behaviors detected by Theorem 1 with N = 0, with a quite fast convergence to the equilibrium. The illustration of the second case Figure 2(b) is consistent with Figure 1, since the solution of this **system** diverges. This is consistent with the fact that no solutions to the conditions of Theorem 1 can be found for any N ≤ 12. More interestingly, the last situation, presented in Figure 2(c), shows simulations results which are very slowly converging to the origin, with however a lightly damped oscillatory behavior of the state of the ODE **and** of the PDE close to the boundary x = 0. On the other side, the state function u(x, t) for sufficiently large values of x is clearly smooth **and** converges slowly to the origin. Actually, case (c) illustrates a situation where a very small diffusion coefficient γ induces a slow convergent behavior for which the conditions of Theorem 1 are only fulfilled for a large parameter N ≥ 5. This may indicate a

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Index Terms—Lyapunov theory, nonlinear systems, power systems **analysis**, region of attraction, sum of squares.
I. I NTRODUCTION
Transient **stability** is one of the most important power systems **analysis** problems. From a physical viewpoint, transient **stability** can be defined as the ability of a power **system** to maintain its synchronism when subjected to large **and** transient disturbances [1]. Widespread assessment tools are generally based on indirect methods which rely on the numerical integration of nonlinear differential equations describing **system** dynamics. However, this approach is not suited to synthesize controllers with a direct quantification of **stability** margin since it does not provide an analytical characterization of **stability** [2]. Alternatively, direct methods are based on the estimation of the **stability** domain of the equilibrium point; they ensure that all the trajectories initiated in this domain converge to the equilibrium point [2]. The main drawback of direct methods is that they rely on the identification of Lyapunov functions which are hard to determine. Indeed, there is not a systematic method for the construction of Lyapunov functions. Moreover, it has been shown that quadratic (i.e., energy type) Lyapunov functions

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