Hybrid & Impulsive Systems

Looped-functionals are a particular class of functionals that express discrete-time stability criteria in an alternative way. The term looped comes from a boundary condition that loops both sides of the functional together. These functionals are useful for several reasons.

A weaker, more general criterion

The first reason is that we work with a discrete-time stability criterion, which is much weaker than its continuous-time counterpart. Asking for a strict continuous decrease of a Lyapunov function is a much stronger requirement than asking for pointwise decrease across a sequence of points extracted from that same function. This is particularly attractive for hybrid systems with jumps in the Lyapunov function level.

Linear impulsive systems are written as

$$ \begin{aligned} \dot{x}(t) &= A x(t), & t \ne t_k \\ x(t_k^+) &= J x(t_k), & k \geq 1 \\ x(t_0) &= x_0 \end{aligned} $$

where $\{t_k\}_{k \ge 0}$ is an increasing sequence of impulse instants with $t_k \to \infty$. Linear switched systems are written as

$$ \dot{x}(t) = A_{\sigma(t)}\, x(t) $$

with $\sigma(t) \in \{1, \dots, N\}$ selecting the active mode.

Convexity, and what it buys us

The stability conditions depend convexly on the system matrices. There is no need to consider the discrete-time system embedded in the hybrid system, which is particularly useful for time-varying and nonlinear systems where no closed-form expression for the embedded discrete-time system exists. Convexity also makes it easy to extend the conditions to uncertain systems.

Looped-functionals act as a unifying paradigm for hybrid systems, including switched and impulsive systems, sampled-data systems, periodic systems, LPV systems, and possibly others. The paradigm is recent, and a great deal of foundational work remains.

Related publications

For the full list, see the publications page (search for "hybrid", "impulsive", "switched", or "sampled-data").