Exploiting isochrony in self-triggered control

Event-triggered control and self-triggered control have been recently proposed as new implementation paradigms that reduce resource usage for control systems. In self-triggered control, the controller is augmented with the computation of the next time instant at which the feedback control law is to be recomputed. Since these execution instants are obtained as a function of the plant state, we effectively close the loop only when it is required to maintain the desired performance, thereby greatly reducing the resources required for control. In this paper we present a new technique for the computation of the execution instants by exploiting the concept of isochronous manifolds, also introduced in this paper. While our previous results showed how homogeneity can be used to compute the execution instants along some directions in the state space, the concept of isochrony allows us to compute the executions instants along every direction in the state space. Moreover, we also show in this paper how to homogenize smooth control systems thus making our results applicable to any smooth control system. The benefits of the proposed approach with respect to existing techniques are analyzed in two examples.

💡 Research Summary

The paper addresses a central challenge in self‑triggered control: determining the next update instant in a way that maximizes resource savings while guaranteeing the desired closed‑loop performance. Existing self‑triggered schemes rely heavily on system homogeneity; they can compute inter‑execution times analytically only along specific directions (typically radial lines emanating from the origin). Consequently, for general smooth nonlinear systems the computed sampling intervals become overly conservative in many directions, limiting the potential reduction in communication and computation.

To overcome this limitation, the authors introduce the notion of isochronous manifolds. An isochronous manifold is defined as the set of states that share the same remaining time until the next control update. By constructing such a manifold, the controller can guarantee that, regardless of the direction in which the state evolves, the time to the next trigger is identical for all points on the manifold. This geometric object generalizes the scaling property of homogeneous systems to the whole state space, thereby eliminating direction‑dependent variability.

A key technical contribution is a systematic homogenization procedure that transforms any smooth control system (\dot{x}=f(x,u)) into a homogeneous one. The authors augment the original state with an auxiliary scalar variable and introduce a positive scaling function (\alpha(\cdot)) that yields a new state vector (\tilde{x}) and dynamics (\dot{\tilde{x}}=\alpha(\tilde{x})\tilde{f}(\tilde{x},\tilde{u})) possessing a well‑defined homogeneity degree (p). This transformation preserves the original system’s trajectories up to a time re‑parameterization, allowing the rich toolbox of homogeneous analysis to be applied to otherwise non‑homogeneous plants.

Once the system is homogeneous, the isochronous manifold (\mathcal{M}) can be derived analytically or approximated numerically. The distance from the current state (x(t)) to (\mathcal{M}) is measured using the inner product between the manifold’s normal vector (n(x)) and the vector field (f(x)). The rate of change of this distance is (\dot{d}=n(x)^{\top}f(x)). The next triggering instant is then computed as

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