Dynamic behavior of the discrete‐time double integrator with saturated locally stabilizing linear state feedback laws

Abstract

SUMMARYThe main contribution of this paper is to completely characterize the dynamic behavior of the discrete‐time double integrator with a saturated locally stabilizing linear state feedback law. In continuous‐time setting, any linear state feedback control law that locally stabilizes the double integrator also globally stabilizes the system in the presence of actuator saturation. In discrete‐time setting, the equivalent of the double integrator does not have the same property. In this paper, we completely characterize the global behavior of saturated locally stabilizing linear state feedback laws for the discrete‐time double integrator. Copyright © 2012 John Wiley & Sons, Ltd.

Synchronization for heterogeneous networks of introspective right-invertible agents with uniform constant communication delay

“…The next lemma is shown in [36]: Lemma 6: For a given setḠβ ,γ,φ andτ > 0, let the conditions in Theorem 2 be satisfied. There exists an ε * such that for ε ∈ (0, ε * ], the system (20) with (19) is globally asymptotically stable for |λ | ∈ (β ,γ), arg(λ ) ∈ [−φ,φ] and τ ∈ [0,τ].…”

“…For heterogeneous networks, the notion of state synchronization may no longer make sense as each agent has a state which may be inherently different from others. In this case, it is more natural to study an alternative problem of output synchronization, that is, all the agents should agree on a set of pre-selected outputs (see, for example, [15]- [20]). In this body of work, it is commonly assumed that the agents are introspective, that is they have a local measurement of their own states besides what is received from the network.…”

Second-order counterexamples to the discrete-time Kalman conjecture

“…The least integer n is called the period of the solution. It is standard to show that the origin is not GAS by finding a periodic solution with period larger than 1 (see Hu & Lin, 2001;Yang et al, 2013).…”

“…Periodic solutions to Lur'e systems where the LTI system is second-order discrete-time but open-loop unstable have been discussed elsewhere in the literature (e.g. Hu & Lin, 2001;Yang, Stoorvogel, & Saberi, 2013); since the LTI system is openloop unstable these cannot be considered counterexamples to the Kalman conjecture.…”

Event‐based global stabilization of linear systems via a saturated linear controller

“…When a system contains no strictly unstable modes, it can be globally stabilized in spite of the input saturation . Furthermore, if every marginally unstable eigenvalue, except the ones corresponding to double‐integrator dynamics, of the system is semi‐simple, linear control law can be used for global stabilization . In addition, linear controller works for semi‐global stabilization as well if the initial value of the marginally stable system is known to be located within a bounded set .…”