Sampled-data control under magnitude and rate saturating actuators: SAMPLE-DATA CONTROL UNDER SATURATING ACTUATORS

[14, Th. 3] achieves η2 = 0.176s for the same time-invariant case.

“Conversely, [14, Th. 3] achieves η 2 = 0 . 176s”

Lemma 2: Under the stated conditions, any initial condition x(0) belonging to the set L_V ensures that (i) the increment of the Lyapunov function V between two consecutive sampling instants is strictly negative, and (ii) the trajectories of the closed-loop system (6) with aperiodic samplings η_k ∈ [η1, η2] converge asymptotically to the origin without leaving L_V.

“Then, any initial condition x ( 0 ) belonging to the set:”

Remark 1: The definition in (10) is necessary to ensure that the actuator state trajectories remain within S_R(μ_R) throughout the sampling intervals.

“Following the approach in [14], where rate saturation is considered for LTI systems, the definition in (10) is necessary to ensure that the actuator state trajectories remain within S R (μ R ) throughout the sampling intervals.”

Conditions in [14] cannot be used for the LPV system in Example 2.

“Note that conditions in [14] cannot be used in this case due to the LPV system.”

The use of model (2) to describe magnitude and rate saturating actuators can be seen as a particular case of nested saturation [16].

“As stated by [15] and [14], the use of model (2) to describe magnitude and rate saturating actuators can be seen as a particular case of nested saturation [16].”

The proposed method yields less conservative stability conditions compared to existing approaches.

“Two numerical examples demonstrate the effectiveness of the proposed method, highlighting less conservative stability conditions compared to existing approaches.”

For the time-invariant case with ρ(t)=0, using O1 with R0=1.5I and parameter-independent matrix variables, the upper limit of the sampling interval is η2 = 0.205s.

“To compare our conditions with [14], we consider the timeinvariant case with ρ( t ) = 0. Using the optimization procedure O 1 with R 0 = 1 . 5 I , assuming all matrix variables are parameter-independent, we find an upper limit of the sampling interval of η 2 = 0 . 205s.”

Lemma 1: If x( ̃t) ∈ S_M(μ_M) and (x(t), x( ̃t), Ψ_M) ∈ S_R(μ_R), then the dead-zone functions Ψ_M and Ψ_R satisfy the inequalities (8) and (9) with diagonal matrices 0 < T_M ∈ S^nu and 0 < T_R ∈ S^nu.

“If x ( ˜ t ) ∈ S M (μ M ) and ( x ( t ), x ( ˜ t ), Psi1 M ) ∈ S R (μ R ) , then the dead-zone functions Psi1 M and Psi1 R , satisfy”