State-dependent sampling for Linear Time Invariant systems: A discrete time analysis

Abstract

This work concerns the adaptation of sampling times for Linear Time Invariant (LTI) systems controlled by state feedback. Complementary to various works that guarantee stabilization independently of changes in the sampling rate, here we provide conditions to design stabilizing sequences of sampling instants. In order to reduce the number of these sampling instants, a dynamic scheduling algorithm optimizes, over a given sampling horizon, a sampling sequence depending on the system state value. Our proofs are inspired by switched system techniques combining Lyapunov functions and LMI optimization. To show the applicability of the technique, the theoretical study is illustrated by an implementation in Matlab/TRUE TIME.

Recent developments on the stability of systems with aperiodic sampling: An overview

“…Self-triggered control mechanisms with stability guarantees have been proposed in [264,265,6,64,23] using the Input/Output stability approach, in [166,161,255] using discrete-time Lyapunov functions, in [61,63] using convex embeddings, in [219] using a hybrid formulation and in [62] using a time-delay system approach.…”

Stability of discontinuous homogeneous nonlinear sampled-data systems

“…in order to use well known results about linear systems having a constant sampling period. Indeed, we know that the closed-loop linear sampled-data system (9) under the linear controller u(x(t k )) = −k 1 x 1 (t k )−k 2 x 2 (t k ) with the samplings (20) is asymptotically stable if and only if the matrix Λ(T ) of its associated linear difference equation is Schur [34,35]. For the double integrator with k 1 = 5 and k 2 = 2.5, Λ(T ) is Schur if and only if T < T Schur = 0.8s.…”

Stability of homogeneous nonlinear systems with sampled-data inputs

“…• If we select α = 1 then the closed-loop system ( 16)-( 17)-( 18) is linear. In this case, we know that the linear closed-loop sampled-data system ( 16)-( 17)-( 18) is asymptotically stable if and only if the matrix Λ(T ) of the linear difference equation associated with ( 16)- (17) and defined in [6,35] is Schur. For the double integrator, Λ(T ) is Schur if and only if T < T Schur = 2s.…”