Stability of LTI Systems with Distributed Sensors and Aperiodic Sampling

Abstract

This chapter is dedicated to the stability analysis of sampled-data linear time invariant systems with asynchronous sensors and aperiodic sampling. The study is performed using an input/output interconnection modelling, and tools from the robust control theory. Two approaches are presented. One is based on the small gain theorem, while the other is based on the dissipativity theory. Tractable stability criteria that allow an estimation of the Maximal Admissible Sampling Period are obtained for both approaches. Finally, experimental results performed on an inverted pendulum benchmark are presented. They confirm the applicability of both approaches and allow for some comparisons between both results.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

“…with f ∈ L 2e [0, ∞). In the following theorem, we provide conditions under which the single-loop sampleddata LTI system (2)-(3) can be represented using the feedback-interconnection of operators G and ∆, given by (7) and (9), respectively. This theorem builds upon similar transformations given in [3,6,11,14,15,16,19].…”

“…Additionally, in this formulation, non-linearities can be treated as operators represented by perturbations. In the second type of formulation, state-space model representations with non-zero initial conditions are considered [20,7]. The robustness with respect to asynchrony is then given in terms of exponential stability property.…”

${L}_{2}$-Stability Criterion for Systems with Decentralized Asynchronous Controllers

“…Considering the rectangular signalx i (t), representing the sampled version of system state x i (t), and the signalx i (t) as given in (12) representing the signal actually used at the level of actuators, the effect of asynchrony can be captured by introducing an error…”

“…However, the period between sampling and actuation instants was treated to be constant and all the system state data was considered to be sampled at same time instants. In the scope of sampled-data and networked control systems [10,11,12], there are a few similarities with the problem we consider. For example, decentralized event-triggered control and delay introduced due to network effects can also be seen as a form of asynchronism [13].…”

Investigating new classes of sampling sequences: Application to the stability analysis of decentralized sampled-data systems

“…Using the same hybrid framework as [11], [12] shows stability of unsynchronized saturated linear systems. In [13] and [14] small gain arguments are used to establish stability of decentralized sampled-data systems while [15] use integral quadratic constraints to achieve the same objective.…”