Sampled-Data Control of LPV Systems With Magnitude and Rate Saturating Actuators

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

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

• cites [14] sourceNot checked — no full text of the source

In Example 2, when LPV parameters are available with | ̇q1|≤0.15, | ̇q2|≤0.8, and | ̇q3|≤1.7, Theorem 1 yields a maximum η2 = 0.114s.

“In case the LPV parameters are available with |˙ q 1 | ≤ 0 . 15, |˙ q 2 | ≤ 0 . 8, and |˙ q 3 | ≤ 1 . 7, Theorem 1 yields a maximum η 2 = 0 . 114s.”

For Example 2 with O1, the maximum sampling interval bound is η2 = 0.101s.

“With O 1, we reach η 2 = 0 . 101s as the maximum sampling interval bound.”

The proposed method is effective, as demonstrated by two numerical examples.

“Two numerical examples demonstrate the effectiveness of the proposed method”

• Also stated in: § VI. CONCLUSION · p. 7 · boosted

  “Two numerical examples have demonstrated the efficiency of the proposed design”

Lemma 4: If α_i(t) ∈ C^1 and | ̇α_i(t)| ≤ φ_i for i=1,...,N, then it is equivalent to rewrite Y_ ̃α = Y_α - τ(t) ̇Y_c.

“If α i ( t ) ∈ C 1 and | ˙ α i ( t ) | ≤ φ i , for i = 1 , . . . , N , thus it is equivalent to rewrite”

• cites [19] Oliveira, Refining parameter mismatch handling in LPV sampled-data controller design, AutomaticaNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The time-varying parameter α_i(t) is continuously differentiable and its derivative is bounded by φ_i: α_i(t) ∈ C^1 and |˙α_i(t)| ≤ φ_i for all t ≥ 0 and i=1,...,N.

For Example 1 with η2=0.1s and κ=1.5, applying O2 yields γ=55.85 and controller gains K1=[-0.0078 -0.0084 0.9065] and K2=[-0.1941 -0.0875 0.7128].

“Considering η 2 = 0 . 1s and applying the optimization procedure O 2 with κ = 1 . 5, we obtain γ = 55 . 85, controller gains K 1 = [ -0 . 0078 -0 . 0084 0 . 9065 ] and K 2 = [ -0 . 1941 -0 . 0875 0 . 7128 ]”

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”

• cites [14] sourceNot checked — no full text of the source

Theorem 1: Suppose there exists matrices ... such that ... Then, for any initial condition x(0) belonging to E(P_α) = {x ∈ R^nx, x^⊤ P_α x ≤ 1}, where P_α = ̃X^{-⊤} P_α ̃X^{-1}, the respective trajectories of the LPV sampled-data system (6) with K_ ̃α = ̃K_ ̃α ̃X^{-1} converge asymptotically to the origin.

“Then, for any initial condition x ( 0 ) belonging to the set E ( P α) = { x ∈ R nx , x ⊤ P α x ≤ 1 } , where P α = ˜ X -⊤ P α ˜ X -1 , the respective trajectories of the LPV sampled-data system (6) with K ˜ α = ˜ K ˜ α ˜ X -1 converge asymptotically to the origin.”

• AssumptionThe authors state this as an assumption: The time-varying parameter α_i(t) is continuously differentiable and its derivative is bounded by φ_i: α_i(t) ∈ C^1 and |˙α_i(t)| ≤ φ_i for all t ≥ 0 and i=1,...,N.
• AssumptionThe authors state this as an assumption: The time-varying parameter vector α(t) is online available.
• AssumptionThe authors state this as an assumption: The sampling interval η_k varies but remains constrained by two positive scalars η1 ≤ η2, such that 0 < η1 ≤ η_k ≤ η2.
• AssumptionThe authors state this as an assumption: The matrices A(α(t)) and B(α(t)) have a polytopic structure with N known vertices.

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.”

• Also stated in: § V. ILLUSTRATIVE EXAMPLES · p. 6 · plain

  “Conversely, [14, Th. 3] achieves η 2 = 0 . 176s, highlighting that the proposed functional in this letter leads to less conservative solutions.”

• cites [14] sourceNot checked — no full text of the source

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.”

• cites [14] sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: In the comparison of Example 1, all matrix variables are parameter-independent.

In Example 2, the increase in the safe ball from robust to LPV case occurs at the expense of increasing decision variable numbers from V1=1275 to V8=9521.

“Such an increase in the safe ball occurs at the expense of increasing the decision variable numbers from V 1 = 1275 (robust gain) to V 8 = 9521 (LPV gain).”

Lemma 3: If there exists a matrix M_α ∈ R^{p×p} satisfying M_α + M_α^⊤ ≥ 0 such that (15) holds, then f(y) < 0 for all y ∈ [h, h].

“If there exists a matrix M α ∈ R p × p satisfying M α + M ⊤ α ≥ 0 , such that”

• Also stated in: § III. AUXILIARY RESULTS · p. 5 · plain

  “then it follows that f ( y ) &lt; 0, ∀ y ∈ [ h , h ].”

• cites [17] sourceNot checked — no full text of the source

From (15), the level set E(P_α) can be exactly computed as the intersection of a finite number of ellipsoids, given by ⋂_{α(t)∈Γ} E(P_α) = ⋂_{i∈1,...,N} E(P_i).

“From (15), the level set E ( P α) can be exactly computed as the intersection of a finite number of ellipsoids, given by ⋂ α( t ) ∈ Gamma1 E ( P α) = ⋂ i ∈ 1 ,..., N E ( Pi )”

In Example 1, when ρ(t) is not available for feedback, the robust controller achieves a maximum feasible limit η2 = 0.078s.

“A simpler controller implementation, for instance when ρ( t ) is not available for feedback, can be provided with a robust controller following the directions in Remark 2, achieving more conservative results. In this case, the maximum feasible limit is η 2 = 0 . 078s.”

Modeling the parameter mismatch between the continuous-time system and the sampled controller using the mean value theorem leads to less conservative conditions.

“The mismatch between the continuous-time system's parameters and the sampled controller's parameters is modeled using the mean value theorem, which leads to less conservative conditions.”

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.”

• cites [14] sourceNot checked — no full text of the source

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.”

• cites [14] sourceNot checked — no full text of the source

Remark 2: When the parameters α(t) are not available for feedback, robust controllers can be obtained by setting ̃K_α = ̃K and ̇ ̃K_c = 0, and all parameter-dependent matrices Y_α = Y, yielding the corresponding derivative ones as zero in Theorem 1.

“When the parameters α( t ) are not available for feedback, we can obtain robust controllers by setting ˜ K α = ˜ K and ˙ ˜ K c = 0 , and all parameter-dependent matrices Y α = Y , yielding the corresponding derivative ones as zero in Theorem 1.”

The case of LPV systems with saturating actuators in both magnitude and rate has not yet been investigated in sampled-data control systems.

“Thus, to the best of the authors' knowledge, the case of LPV systems with saturating actuators in both magnitude and rate has not yet been investigated in sampled-data control systems.”

The proposed conditions can be incorporated into standard optimization procedures to maximize the sampling interval or the region of attraction.

“The proposed conditions can be incorporated into standard optimization procedures to maximize the sampling interval or the region of attraction within which the trajectories asymptotically converge to the origin.”

The proposed conditions can be simplified to recover a robust controller design whenever the time-varying parameter is unavailable.

“The proposed conditions can be simplified to recover a robust controller design whenever the time-varying parameter is unavailable.”

• Also stated in: § IV. MAIN RESULTS · p. 6 · plain

  “When the parameters α( t ) are not available for feedback, we can obtain robust controllers by setting ˜ K α = ˜ K and ˙ ˜ K c = 0 , and all parameter-dependent matrices Y α = Y , yielding the corresponding derivative ones as zero in Theorem 1.”