Here, the considered paradigm is the one developed in chapter 2 which associates to a continuous-time flow whose model is of a differential equation type, a discrete- time behavior capturing the switched characteristic imposed by the **control** law. Two subsets included in an extended space, built from the state and the **control** spaces, determine the regions where the continuous and discrete dynamics are active. The main tool for proving stability of a compact attractor defined in this extended space, proceeds by an appropriate extension of Lyapunov stability theory developed in the context of **hybrid** dynamic systems in [26] and [34]. Due to the affine structure of the modes, a quadratic Lyapunov function can be selected from a positive definite symmetric matrix satisfying a set of Lyapunov inequalities. A **hybrid** **control** law with its two associated flow and jump is deduced from this matrix and an upper bound of a LQ performance index for the controlled system can be computed. It is possible to deduce an optimal guaranteed cost **control** law leading to the tight upper bound, by solving a LMI optimization problem.

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of Dynamic Programming techniques. As a general analytical background, the reader is referred to [4, 5, 6] for a study of the characterization and properties of the value function.
The contribution of the present paper is on the numerical side. Our first point of interest is to approximate the value function of the problem – in this part, construction of a convergent approximation will result from an adaptation of monotone schemes to the **hybrid** case, which involves a dynamic programming equation in the form of a quasi-variational inequality. Moreover, in order to give a more constructive result, we also study the synthesis of an approximate optimal feedback **control**. We will define a procedure to compute a piecewise constant **control** given the state and the numerical value function, and prove that (under suitable assumptions) this construction provides an asymptotically optimal solution. Note that, in general, the value function of most relevant **hybrid** **control** problems is not expected to be continuous unless a certain number of technical assumptions are satisfied. In what follows, we will keep ourselves within this more restrictive framework, but, as numerical examples will show, the approximation strategies under consideration are robust enough to provide good results even in more general situations.

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concernant des méthodes numériques pour le contrôle optimal de systèmes hybrides, qui peuvent être contrôlés simultanément par des fonctions mesurables et des sauts discontinus dans la variable d'état. La deuxième partie est dédiée à l'étude d'une application spécique sur l'optimisation de trajectoires pour des lanceurs spatiaux avec contraintes en probabilité. Ici, on utilise des méthodes d'optimisation nonlineaires couplées avec des techniques de statistique non paramétrique. Le problème traité dans cette partie appartient à la famille des problèmes d'optimisation stochastique et il comporte la minimisation d'une fonction de coût en présence d'une contrainte qui doit être satisfaite dans les limites d'un seuil de probabilité souhaité. Title : Numerical methods for **hybrid** **control** and chance-constrained optimization problems

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nected in cascade, which requires two switching, increasing the commutation losses. In the QBC, one can obtain the same gain using only one conversion stage increasing the efficiency.
The novelty of this paper is to integrate a state observer in a **hybrid** **control** scheme of the converter to avoid using cur- rent sensors. To the best of Authors’ knowledge, an observer has not been designed before for a switched a ffine system con- trolled with a Lyapunov matrix-based min-projection **control**, in a **hybrid** dynamical model. This fact is particularly important because it allows to remove current sensors. From a practical point of view, the lack of current sensors can improve the pro- duction cost of converters. Indeed, if currents are not measured, it is not needed to employ neither current sensors, nor Analog to Digital Converters (ADC) any more; and the dimensions of the board could be better optimized. Moreover, the fact of using a QBC, which is four dimensions, justifies the use of an observer to reduce the number of sensors. The stability of the complete system is established in terms of Uniform Global Asymptotic Stability (UGAS) of a bounded and closed set that includes the operating point, providing UGAS of both tracking error of the equilibrium point and estimation error. It is useful to point out that the property of UGAS of such set implies the robustness of the complete **control** loop with respect to small perturbations around the operating point.

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have their performance rapidly decreasing with time compared to pure centralized ones (or predictive ones) if no perturbation occurs (that is, all data are known and certain from the beginning).
Due to the limitations of these two historical approaches facing the current industrial needs, researchers are more and more considering a third kind of approach by trying to propose integrated scheduling and **control** architectures that integrate local distributed-reactive mechanisms implemented into products/resources **control** holons/agents with global centralized scheduling mechanisms, being robust or not. Such architectures named **hybrid** scheduling and **control** architectures (in short, HCA for **Hybrid** **Control** Architectures in the remaining of this document) are intended to capitalize the

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4. An **hybrid** **control** Scheme
The inverter model presented in (2) covers two different types of dynamics.
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In one hand, the continuous-time variables which are the differences and the sums of the inductor currents and the capacitor voltages. In other hand, the **control** input, v, is a discrete-time variable. Consequently, the framework given in [12] about **hybrid** dynamical system is well suited.

35 35.2 35.4 35.6 35.8 36 36.2 36.4 36.6 36.8
5 Conclusion and future works
We applied **hybrid** **control** architecture on a mobile robot to obtain a stable trajectory tracking while avoiding obstacles. Indeed, the robot is controlled by elementary continuous controllers according to the sub-tasks to accomplish (trajectory tracking, obstacle avoidance) and switching from a controller to an other is done referring to discrete events. We saw that hard switches cause chattering and are not efficient to insure a safe navigation. Therefore, we propose to use the multiple Lyapunov function for **hybrid** systems to design a stable **hybrid** **control** architecture. In addition to elementary stable controllers for the two main sub-tasks, we introduce a third controller which insures the second sufficient condition of multiple Lyapunov function (cf. section 2). Simulations show that our architecture prevents useless switches, guaranteeing thus a safe navigation for the robot. Applying this stable **control** architecture to a dynamic environment (e.g., moving obstacles), will be the subject of future works. Application to multi-robot systems navigating in formation seems also to be interesting. The objective is to make each robot able to avoid an obstacle before regaining the convoy.

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In the second subspace, called X2, the master robot is position controlled, and the force exerted by the operator are used as input commands for the slave robot [r]

and veri…es the resulting controller design. The approach is to harness the power of industrially proven system modeling (simulation tools). The nonlinear continuous models are wrapped in a discrete abstraction layer that is based on event detection. The discrete abstraction combined with a DES speci…cation in a limited horizon reachabil- ity computation, produces a discrete event (DE) controller that is, within this limited space and time, locally safe. Furthermore, a reduction in computational complexity is achieved by exploit- ing a lazy (just-in-time) synchronous composition of the speci…cation and plant models at design time. This scheme is implemented as an online computation, in order for the controller operation to be extended into an in…nite time horizon. Limited lookahead (LL) supervisory **control** has been extensively studied in a DES setting by (Chung et al. 1992). In (Raisch and O’Young 1998), discrete abstractions based on the trun- cated time history of discrete-time LTI continu- ous models were used to synthesize DES super- visory controllers. Others, (Su et al. 2003) and (Abdelwahed et al. 2005), have also used discrete abstractions of switched continuous systems in a LL framework to e¤ect **control** over **hybrid** sys- tems. Similar to our work is (Stursberg 2004), in which the nonlinear continuous dynamics are retained as embedded simulations, and a graph search algorithm has been described for optimal **hybrid** **control**. Our approach di¤ers in that we do not limit switching to discrete time intervals and controller graph pruning is done in a maximally permissive sense with respect to safety as is com- monly done in optimal DES supervisory **control** (Ramadge and Wonham 1987).

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24 Rue Alain Savary, 25000 BESANCON, France {Lounis.Adouane & Nadine.Piat}@ens2m.fr
Abstract - This paper presents a **hybrid** **control** architecture
based on subsumption and schemas motors principles in order to achieve complex and cooperative tasks. The **control** architecture implemented is constituted by a set of independent and elementary behaviors organized in layers of skills. Specific low-level behaviors, called altruistic behaviors and inspired by societies of insects (attractive or repulsive signals), are used to improve the efficiency of the **control**. Therefore, competitive and cooperative mechanisms are used in an unique **hybrid** architecture of **control** to perform a complex box-pushing task by a set of mini-robots. The analysis of an elevated number of simulations allows us to have statistical results (time to complete the task was chosen as performance criteria) which show the existence of an optimal number of robots to achieve the box-pushing task and underline the importance of the use of altruistic behaviors to enhance the cooperative task. Index Terms - Cooperative robotics, Behavioral architecture of **control**, Altruistic behavior, Box-pushing task.

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Alternatively, core material systems can be found in entangled fibers [ 14 , 15 ], or as viscoelastic inclusions embedded in elastic layers [ 16 ]. Unfortunately, this technique does not always allow effective passive damping. To improve the damping properties, it is sometimes necessary to increase the thickness of the viscoelastic layer. However, limitations are quickly encountered in terms of dimensions and mechanical properties of composites, which become less resistant as the thickness of the viscoelastic layers becomes larger. Other limitations are sometimes associated with the total thickness of the structure, which is related to its practical implementation. To overcome these restrictions while increasing the damping properties of multilayer structures, the use of piezoelectric materials is desirable and sometimes even unavoidable. Indeed, the attractive properties of piezoelectric materials, and especially their ability to deform under electrical load and vice versa, make them indispensable. More specifically, the current generated by deformation is collected and amplified or attenuated in order to **control** the vibration amplitudes. Some literature studies effectively combined passive damping and active **control** using viscoelastic core sandwiches constrained by layers of piezoelectric materials [ 17 , 18 ]. Controlling this phenomenon is therefore important to better implement these **hybrid** **control** systems. In the literature, a number of works have been devoted to this issue, among which the contributions reported in references [ 19 , 20 ]. In particular, it is clearly shown in these works that the finite element discretization leads to frequency-dependent nonlinear problems. The complexity of such problems restricted the earlier investigations to PD (Proportional Derivative) type of **control** laws (see Duigou- Kersulec [ 19 ] and Boudaoud [ 20 ], among others).

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Abstract— This paper deals with a multi-mode **control** ar-
chitecture for robot navigation while using **hybrid** **control**. It presents, an adaptive and flexible mechanism of **control** which guarantees the stability and the smoothness of the switch between controllers. Moreover, a specific safety mode is proposed and applied on the robot which navigates very close to obstacles. The overall architecture allows to obtain very smooth trajec- tories while guaranteeing very safe obstacle avoidance. Many simulations on different robot configurations and cluttered envi- ronments permits to confirm the reliability and the robustness of the proposed **control** architecture. In addition, an appropriate indicator is proposed to quantify the trajectory smoothness.

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In the last two decades, the concept of **hybrid** **control** system has provided a sound math- ematical framework for treating **control** systems in which continuous and discrete **control** actions mix together, and this framework has also been successfully adapted to optimal **control** problems. Among the various systems covered by this theory, we mention economic models with restocking, multigear and **hybrid** vehicles, and, more in general, systems with switchings in the dynamics and/or impulsive changes in the state. In this paper, we study efficient numerical methods for applying Dynamic Programming techniques to **hybrid** op- timal **control** problems of infinite horizon type.

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For HSPGS, which are based on solar PV array and DG, two different topologies are selected for study. The first one uses DE driven fixed speed SG. Dump load and BESS are tied to the DC bus instead of AC side. To get MPPT from solar PV array P&O technique is used. AC voltage and the system frequency regulation, as well as, the power quality improvement at the PCC, are achieved by using the modified instantaneous p-q theory **control** algorithm. To protect the BESS from the overcharging, dump load is controlled. To optimise fuel consumption by DG and to improve its efficiency, a new controlled switch is added between the PCC and the terminals of DG in order to isolate it from the rest of installation to behave as backup energy source during the day time. The second selected HSPGS uses DE driven variable speed DFIG, the solar PV array is tied to the PCC through two-stage inverters and the rotor of the DFIG is connected to DC bus using AC/DC converter. To achieve MPPT from the solar PV array P&O technique and to regulate the AC voltage and the system frequency, as well as, to improve the power quality at the PCC, indirect **control** technique is used. To regulate the rotational speed of the DG according to the load power demand, AC/DC converter is controlled using stator flux oriented **control** technique when the reference DG rotational speed is estimated using reel characteristic of optimal rotational speed of DG versus load curve.

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V. C ONCLUSION
A study of force feedback based guiding task is proposed in this paper. The lateral contact force is estimated by using two sensorized fingers and is used in an external **hybrid** force/position **control**. A simulator development based on Matlab-Simulink model is investigated and the use of this simulator enables to set parameters of the controller. It prevents for breaking a part of the system if the **control** is unstable. The effectiveness of the incremental controller is validated and it was shown that the time for canceling a step perturbation can reach the corrector stage dynamic ( ≈ 35 ms). This controller depends on sampling frequency and step increment: the magnitude of this step has to be smaller than the play for ensuring the stability. An experimental setup is developed to perform the guiding task. A micropart of 2 mm x 50 µm x 50 µm in size is grasped, the incremental controller is validated by rejecting perturbations at each side of the rail. A ramp tracking with 2 µm/s slope which

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Fig. 3 Virtual **hybrid** test results for a more complex imposed crack path: (a) imposed cracked path, (b) X-FEM results, (c) damage model crack path confronted with the desired trajectory. 4. Conclusions
By coupling a LEFM model with a damage model a **hybrid** virtual test was developed. Considering the complexity of a real **hybrid** test, the use of such numerical simulations is very important to prepare, to anticipate difficulties that may arise during the real test. In perspective, in order to have a more accurate and a more reactive response, the numerical **hybrid** test will be improved and optimized.

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? This research is part of the research programme High Tech Systems and Materials (HTSM), which is supported by NWO domain Applied and
Engineering Sciences and partly funded by the Ministry of Economic Affairs.
1 R. Beerens, M. Heemels, H. Nijmeijer, and N. van de Wouw are with the Department of Mechanical Engineering, Eindhoven University of Technology,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands {r.beerens; w.p.m.h.heemels; h.nijmeijer; n.v.d.wouw;}@tue.nl 2 A. Bisoffi is with the Department of Automatic **Control**, KTH Royal Institute of Technology, 10044 Stockholm, Sweden bisoffi@kth.se 3 L. Zaccarian is with CNRS, LAAS, Univ. de Toulouse, LAAS, F-31400 Toulouse, France, and also with Dip. di Ingegneria Industriale, University of

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