CNRS, ENS Paris-Saclay 94235 Cachan Cedex, France.
Abstract: In this paper, we consider the problem of symbolic model design for the class of incrementally stable **switched** **systems**. Contrarily to the existing results in the literature where switching is considered as periodically controlled, in this paper, we consider aperiodic time sampling resulting either from uncertain or event-based sampling mechanisms. Firstly, we establish sufficient conditions ensuring that usual symbolic models computed using periodic time-sampling remain approximately bisimilar to a **switched** system when the sampling period is uncertain and belongs to a given interval; estimates on the bounds of the interval are provided. Secondly, we propose a new method to compute symbolic models related by feedback refinement relations to incrementally stable **switched** **systems**, using an event-based approximation scheme. For a given precision, these event-based models are guaranteed to enable transitions of shorter duration and are likely to allow for more reactiveness in controller design. Finally, an example is proposed in order to illustrate the proposed results and simulations are performed for a Boost dc-dc converter structure.

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The first category of methods exploits necessary optimality conditions, in the form of Pontryagin’s maximum principle (the so-called indirect approaches), or through a large nonlinear discretization of the problem (the so-called direct approaches). The first con- tributions can be found in [8, 22, 36] where the problem has been formulated and partial solutions provided through generalized Hamilton-Jacobi-Bellman (HJB) equations or in- equalities and convex optimization. In [38, 47, 43], the maximum principle is generalized to the case of general hybrid **systems** with nonlinear dynamics. The case of **switched** **systems** is discussed in [4, 42, 2]. For general nonlinear problems and hence for **switched** **systems**, only local optimality can be guaranteed , even when discretization can be prop- erly controlled [51]. The subject is still largely open and we are far from a complete and numericaly tractable solution to the **switched** optimal control problem.

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Abstract
We study stability criteria for discrete-time **switched** **systems** and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past fifteen years. We prove in this note that a family of language- theoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort.

keywords:Nonlinear control **systems**, reachability, formal methods, numer- ical simulation, control system synthesis.
1 Introduction
In this paper, we present a control synthesis method for a special form of hybrid **systems** [1] named **switched** **systems**. Such **systems** have been recently used in various domains such as automotive industry and, with noteworthy success, power electronics (e.g., power converters). They are continuous-time **systems** with discrete switching events. More precisely, these **systems** are described by piecewise continuous dynamics called modes; the change of modes takes place instantaneously at so-called switching instants. In this paper, we suppose that switching instants occur periodically at constant sampling period τ (sampled 1 Adrien Le Co¨ ent and Laurent Fribourg are at CMLA and LSV, CNRS, ENS Paris-Saclay,

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3, rue Joliot-Curie, 91192 Gif-sur-Yvette, cedex, France
Abstract: Methods for computing approximately bisimilar symbolic models for incrementally stable **switched** **systems** are usually based on discretization of time and space, where the value of time and space sampling parameters must be carefully chosen in order to achieve a desired precision. This often results in symbolic models that have a very large number of transitions, especially when the time sampling, and thus the space sampling parameters are small. In this paper, we present an approach to the computation of symbolic models for **switched** **systems** using multirate time sampling, where the period of symbolic transitions is a multiple of the control (i.e. switching) period. We show that multirate symbolic models are approximately bisimilar to the original incrementally stable **switched** system. The main contribution of the paper is the explicit determination of the optimal sampling ratio between transition and control periods, which minimizes the number of transitions in the symbolic model. Interestingly, this optimal sampling ratio is mainly determined by the state space dimension and the number of modes of the **switched** system. Finally, an illustration of the proposed approach is shown for the boost DC-DC converter, which shows the benefit of multirate symbolic models.

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locally Lipschitz functions that are smooth in some open sets is quite intuitive and well-studied in literature. As a particular example, we refer the reader to (Della Rossa et al., 2018) to see the utility of functions obtained by max- min composition over a finite family of smooth functions in the context of stability of **switched** **systems**.

considered the case where the decay or growth rates of all subsystems are al- lowed to be nonlinear, and the changes of Lyapunov function values at switch- ing instants are also assumed to be nonlinear. Under some mild assumptions on the **switched** system, and using the hybrid system framework, we derived a mixed ADT and AAT condition which guarantees ISS or iISS of the **switched** system. Using the main result of our work, **switched** **systems** with saturating dynamics and **switched** bilinear **systems** were shown to be uniformly iISS over switching signals satisfying this mixed ADT and AAT condition.

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Numerous works have been dedicated to the computa- tion of symbolic models for various classes of dynamical **systems**. Focusing on approximately bisimilar abstrac- tions, existing approaches make it possible to deal with nonlinear **systems** [19,23], **switched** **systems** [10], time- delay **systems** [21,20], networked control **systems** [6,35], stochastic **systems** [32,34]... All these approaches are es- sentially based on discretization of time and space and require the considered system to satisfy some kind of in- cremental stability property [1]. However, incremental stability can be dropped if one seeks symbolic models related only by one-sided approximate simulation rela- tions [28,36]. In most cases, symbolic models of arbitrary precision can be obtained by carefully choosing time and space sampling parameters. However, for a given preci- sion, the choice of a small time sampling parameter im- poses to choose a small space sampling parameter result- ing in symbolic models with a prohibitively large num- ber of transitions. This constitutes a limiting factor of the approach because the size of the symbolic models is crucial for computational efficiency of discrete con- troller synthesis algorithms. Several studies have been conducted in order to address this issue by enabling the computation of more parsimonious symbolic models with smaller numbers of transitions. Such approaches in- clude compositional abstraction schemes where symbolic models of a system are built from symbolic models of its components [30,22,15]; multi-resolution or multi-scale symbolic models computed using non-uniform adaptive space discretizations [31,8]; symbolic models where the set of symbolic states is not given by a discretization of the state-space but by input sequences [12,33].

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Advances of Implicit Description Techniques in Modelling and Control of Switched Systems.. Moisés Bonilla, Michel Malabre, Vadim Azhmyakov.[r]

The anti-windup techniques and the **switched** formalism have been also gathered in the litera- ture. For instance, it is emphasized in (Bruckner et al. 2010) (see also (Bruckner et al. 2013)) that even if the plant does not switch, a switching anti-windup compensator may improve the perfor- mances. Pionner contributions applying (switching) anti-windup techniques on **switched** **systems** have been provided in the literature. In the continuous-time framework several other tools have been considered: multiple Lyapunov functions (Lu and Lin 2009, 2010), piecewise quadratic Lyapunov functions (Mulder and Kothare 2000, Tiwari et al. 2007) or being focused on uncer- tainties. We may mention the following work on a particular class of **switched** **systems** (Shorten et al. 2009).

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In this article, the construction of ISS Lyapunov func- tions is considered for **switched** nonlinear **systems** in cas- cade configuration. The stability analysis for the resulting hybrid **systems** is carried out under an average dwell-time condition on the switching signal, and an asymptotic ratio condition for establishing ISS. The results pave the path for studying the stabilization of **switched** **systems** with dy- namic output feedback. A Lyapunov function similar to the one used for non-sampled ISS system is constructed to design the sampling algorithms and for the analysis of the closed-loop hybrid **systems** with sampled measurements. The results are illustrated with the help of examples and simulations. One of the limitations of the dynamic output feedback problem considered in this paper is that the con- troller requires exact knowledge of the switching signal. It is of interest to develop theoretical tools when there is mis- match in the switching signal between the plant and the controller. One can also consider additional measurement errors, for example, due to quantization of output and in- put in space, as done in (Tanwani et al., 2016). One could also potentially study the affect of random uncertainties, on top of event-based samples, as has been recently pro- posed in (Tanwani and Tell, 2017).

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Starting from a finite family of continuously differentiable positive definite functions, we study con- ditions under which a function obtained by max-min combinations is a Lyapunov function, establishing stability for two kinds of nonlinear dynamical **systems**: a) Differential inclusions where the set-valued right-hand-side comprises the convex hull of a finite number of vector fields, and b) Autonomous **switched** **systems** with a state-dependent switching signal. We investigate generalized notions of directional deriva- tives for these max-min functions, and use them in deriving stability conditions with various degrees of conservatism, where more conservative conditions are numerically more tractable. The proposed con- structions also provide nonconvex Lyapunov functions, which are shown to be useful for **systems** with state-dependent switching that do not admit a convex Lyapunov function. Several examples are included to illustrate the results.

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In the second part of the paper, we propose to use these mul- tiscale symbolic models for the synthesis of safety controllers for **switched** **systems**. For specifications given by safety automata, we introduce the notion of maximal lazy safety controller which exploits the specificities of multiscale symbolic models: it gives priority to transitions of longer durations; the faster transitions and thus the finer scales of the symbolic models are effectively explored only when safety cannot be ensured at the coarser level and fast switching is needed. We present an algorithm computing symbolic models on the fly during controller synthesis, and therefore dynamics at the finest scales are explored only when necessary. We provide experimental results, obtained by the toolbox CoSyMA [12] that show drastic improvements of the complexity of controller synthesis using multiscale models instead of uniform ones defined in [10]. Symbolic controller synthesis algorithms using on the fly computation of uniform symbolic models have also been considered in [13], for specifications described by a target deterministic transition system approximately simulating the observed behavior of the controlled system.

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smaller intervals and zero elsewhere.
3. FIRST CONVERSE LYAPUNOV THEOREM
In this section we establish a first equivalence result for the global uniform ex- ponential stability of a **switched** of the form (9). The crucial step is given by the following lemma, related to the blow-up phenomenon illustrated in Example 1 in the previous section. It is a variant of a result obtained in [23] in the framework of strongly continuous semigroups. While extending the property to **switched** sys- tem of the form (9), the proof given in [23] should be modified in order to replace the semigroup property by (11). We include the modified proof for the sake of completeness.

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In [9], the problem of the classical unidirectional master-slave synchronization has been reformulated from control theory point of view, in terms of (non) linear observer design. The novelty in the present paper is that unlike the above master-slave configuration, here the synchronization is to be achieved bidirectionaly, using two symmetrical observers (one for each subsystem). The idea is to exploit the richness of the nonlinear dynamics: for identical parameters, the system can exhibit qualitatively different behaviors (multistability), such as periodic, chaotic etc according to the initial conditions. This provides the challenging opportunity to analyze the mutual synchronization between two subsystems, which have been tuned to different orbits, but also between subsystems with slightly different parameters. The latter assumption can be used to model physical **systems** subject to ageing, temperature variations etc.

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The technique used in the proof of Theorem 3.1 was generalized in [ 48 ] to the stability analysis of non-autonomous di fference equations of the form ( 1.57), and the corresponding results are presented in Chapter 4. We provide a suitable representation of their solutions in terms of their initial conditions and some time-dependent matrix coefficients, generalizing the technique used in Chapter 3. This enables us to characterize the asymptotic behavior of solutions in terms of such matrix coefficients. In the case of difference equations with arbi- trary switching, we obtain a delay-independent generalization of Hale–Silkowski stability criterion. Using the classical transformations of hyperbolic PDEs into di fference equations, we apply our results to transport and wave propagation on networks, obtaining, as a conse- quence, that exponential stability of such **systems** is robust with respect to variations of the lengths of the network edges preserving their rational dependence structure. We then prove that the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but at most one.

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7 Conclusion
The problem of peak estimation with uncertainty may be upper bounded with an infinite-dimensional LP in occupation measures, and approximated by a sequence of LMIs. Time-independent and time-dependent (including box, and switching) uncertainties are incorporated into this measure framework for continuous and discrete **systems**. Specific application of this method towards the analysis of linear **systems** was discussed. Future work includes performing peak estimation with further specialized uncertainty structures.

been with us at least since the days of the relay. The earliest reference we know of is the work of Witsenhausen from MIT, who formulated a class of hybrid-state continuous-time dynamic **systems** and investigated an optimal control problem [106]. It is worthwhile mention in that there are various models for hybrid **systems**; due to its inherently interdisciplinary nature, the ﬁeld has attracted the attention of people with diverse backgrounds, primarily computer scientists, applied mathematicians, and engineers [54], [99], [55]. However, we consider continuous-time **systems** with discrete switching events, which consist of several subsystems and a switching law that determines the switching times and mode transitions. Such **systems** are called **switched** **systems** and can be viewed as higher-level abstraction of hybrid **systems** [54]. **Switched** system modeling of any real-process dealing with physical variables are in agreement with the time-continuous and uniqueness principle, i.e., the value of every physical variable changes only continuously in time through every intermediate value (initial and ﬁnal), and by possessing a unique value at a speciﬁc time and space. Any synthesized control should be uniquely deﬁned and continuous in time. Recent eﬀorts in **switched** **systems** research have been typically focused on the analysis of dynamic behaviors, such as stability, controllability and observability, and optimal control, among others (e.g., [99], [56], [54]).

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Some results are available in literature for the stability and stabilizability of **switched** hyperbolic **systems** of balance laws or conservation laws. In [16] some sufficient condi- tions for the asymptotic stability are stated, uniformly with respect to a class of switching signals. In [1], a result of stability is given under an arbitrary switching signal using the propagation of the solution along the characteristics. For **switched** **systems** governed by semigroups of linear evolution operators and their stability under an arbitrary switching signal, see [11]. In [8] a star-shaped network with a central node, and the wave equation governing on each edge, is

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