Corentin BriatHybrid dynamical systems: impulsive, switched and sampleddata systemsLoopedfunctionals are a particular class of functionals allowing to express a discretetime stability criteria in an alternative way. The term 'looped’ comes from the fact that the loopedfunctionals satisfy a boundary condition, looping both sides of the functional together. These functionals are particularly useful for several reasons. The first one is the use of a discretetime stability criterion, which is a much weaker condition than a continuoustime condition. Demanding a continuous decrease of a function is indeed a much stronger requirement than asking for a pointwise decrease of a sequence of points extracted from the same function. This hence allows us to relax the constraint on the strict decrease of the continuoustime Lyapunov function. This is particularly interesting when dealing with hybrid systems, such as impulsive or switched systems, where jumps in the Lyapunov function level can occur. Linear impulsive systems are described as where is the state and is a sequence of increasing impulse instants satisfying as . The notation denotes the rightlimit of at . On the other hand, linear switched systems are given by defined as where is the state and the function defines which mode is active at each time . The second one is that the obtained stability conditions depend on the system matrices in a convex way. In this regard, it is not necessary to consider the discretetime system embedded in the hybrid system. This latter fact becomes particularly interesting when timevarying and nonlinear systems are considered since no closedform expression for the embedded discretetime system exists. The convexity of the conditions also permits an easy extension to uncertain systems. Loopedfunctionals can be seen as a unifying paradigm for dealing with hybrid systems, such as switched and impulsive systems, but also sampleddata systems, periodic systems, LPV systems, and possibly many others. This paradigm is quite recent and a lot of things remain to be discovered, notably applications to delay systems, extensions to systems with inputs, loopedfunctionals for control, etc. Clockdependent Lyapunov functions are a particular class of Lyapunov functions that depend on a clock measuring the time elapsed since a particular event such as the last jump of the state for impulsive systems or the last change of mode for switched systems. The obtained criteria are similar to the ones obtained using loopedfunctionals, with the striking difference that they are structurally more suitable for control design. They also involve a much lower number of decision variables, which makes them more scalable as the size of the considered system increases. References:
