Corentin Briat

Hybrid dynamical systems: impulsive, switched and sampled-data systems

Looped-functionals are a particular class of functionals allowing to express a discrete-time stability criteria in an alternative way. The term 'looped’ comes from the fact that the looped-functionals 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 discrete-time stability criterion, which is a much weaker condition than a continuous-time 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 continuous-time 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

begin{array}{rcl} dot{x}(t)&=&Ax(t), tne t_k x(t_k^+)&=&Jx(t_k), kge1 x(t_0) &=&x_0 end{array}

where x is the state and {t_k}_{kge0} is a sequence of increasing impulse instants satisfying t_ktoinfty as ktoinfty. The notation x(t_k^+) denotes the right-limit of x(s) at sdownarrow t_k. On the other hand, linear switched systems are given by defined as


where x is the state and the function sigma(t)in{1,…,N} defines which mode is active at each time t.

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 discrete-time system embedded in the hybrid system. This latter fact becomes particularly interesting when time-varying and nonlinear systems are considered since no closed-form expression for the embedded discrete-time system exists. The convexity of the conditions also permits an easy extension to uncertain systems.

Looped-functionals can be seen as a unifying paradigm for dealing with hybrid systems, such as switched and impulsive systems, but also sampled-data 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, looped-functionals for control, etc.

Clock-dependent 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 looped-functionals, 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.