Semiglobal Switched Predictor-Feedback Stabilization of Nonlinear Systems With Input Delay and State/Input Quantization

The quote only partly states this claim.

Under Assumptions 1 and 2, if ∆ and M satisfy condition (25), then for any fixed μ>0 and E0>0, for all initial conditions satisfying (26), the closed-loop system consisting of plant (2)-(4) and switched predictor-feedback law (17)-(19) has a unique solution and is asymptotically stable.

“Theorem 1: Consider the closed-loop system consisting of the plant (2)-(4) and the switched predictor-feedback law (17)-(19). Under Assumptions 1 and 2, if ∆ and M satisfy”

• Also stated in: § III. STABILITY UNDER STATE QUANTIZATION · p. 5 · plain

  “there exists a unique solution such that X ( t ) ∈ AC ( R + ; R n ) , for each t ∈ R + u ( · , t ) ∈ C rpw ([0 , D ]; R ) , and for each x ∈ [0 , D ] u ( x, · ) ∈ C rpw ( R + ; R ) , which is asymptotically stable.”

• AssumptionThe authors state this as an assumption: System Ẋ = f(X,U) is forward complete.
• AssumptionThe authors state this as an assumption: The system Ẋ = f(X,κ(X)+w) is ISS with respect to w, where κ: R^n→R, satisfying κ(0)=0, is a locally Lipschitz function.
• AssumptionThe authors state this as an assumption: The quantizers q1 and q2 are locally Lipschitz functions satisfying properties P1 and P2 and a saturation level property.

May combine two claims.

The system described by (2)-(4), along with the quantized controller U defined in (17), satisfies the conditions in [19, Theorem 8.1], and the control mapping U(μ,·,·) is locally Lipschitz on bounded sets.

“The system described by (2)-(4), along with the quantized controller U defined in (17), satisfy the conditions in [19, Theorem 8.1].”

• cites [19] [19] I. Karafyllis and M. Krstic, Input-to-State Stability for PDEs, SpringerVerlag, London, 2019. - sourceNot checked — no full text of the source

May combine two claims.

The closed-loop system with state quantization under Assumptions 1 and 2, if ∆ and M satisfy (25), is asymptotically stable for any E0>0 such that E0 ≤ ρ2^{-1}(ρ1(M1)), and the state satisfies |X(t)| + ||w(·,t)||∞ ≤ ρ1(M1) for all t ≥ 0.

“For µ 0 = 1 M ρ -1 1 ( ρ 2 ( E 0 )) , relation (48) and assumption (26) show stability , as well as that | X ( t ) | + ∥ w ( · , t ) ∥ ∞ ≤ ρ 1 ( M 1 ) , t ≥ 0 , for any E 0 > 0 such that E 0 ≤ ρ -1 2 ( ρ 1 ( M 1 )) (for any given M 1 ), which makes Lemmas 1 and 2 legitimate. Asymptotic stability follows from convergence of µ to zero.”

May combine two claims.

The estimates obtained enable establishing that the quantizer does not saturate and that, at each time interval of constant quantization error/range, its error reduces by a certain factor assigned, provided that its nominal range is sufficiently large, also depending on the size of initial conditions.

“The estimates obtained enable to establish that, in fact, the quantizer does not saturate and that, at each time interval of constant quantization error/range, its error reduces (by a certain factor assigned), provided that its nominal range (i.e., for a fixed value of the zoom variable) is sufficiently large, also depending on the size of initial conditions.”

May combine two claims.

The main contributions of the paper are (i) the development of two new, switched nonlinear predictor-feedback laws that simultaneously compensate long input delays and the effect of state/input quantization, achieving semiglobal asymptotic stabilization, and (ii) the introduction of novel stability analysis approaches of the resulting closed-loop systems, together with the study of well-posedness.

“The main contributions of the paper are thus summarized as i) the development of two new, switched nonlinear predictorfeedback laws that simultaneously compensate long input delays and the effect of state/input quantization, achieving semiglobal asymptotic stabilization and ii) the introduction of novel stability analysis approaches of the resulting closed-loop systems, together with the study of”

May combine two claims.

In simulations, for a small fixed value of μ, the system diverges because the quantizer saturates; for a large fixed μ, the states remain bounded, but the large quantization error results in a limit-cycle-type behavior.

“We note that in simulations for a small fixed value of µ , the system diverges because the quantizer saturates. For a large fixed µ , the states remain bounded, but the large quantization error results in a limit-cycle-type behavior.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Because the two control designs are different, the technical details of the stability analysis, the choice of the design parameters of the zoom mechanism μ0, ω, and Tk, and the required stability conditions are also different, although the structure of the proof strategy and the main technical tools are similar.

“As a result, since the two control designs are different, the technical details of the stability analysis, and thus, the choice of the design parameters of the zoom mechanism µ 0 , ω , and T k , as well as the required stability conditions are also different; even though indeed the structure of the proof strategy and the main technical tools employed are similar in the two cases.”

May combine two claims.

For the input quantization case, a Filippov solution P(t) exists and is absolutely continuous on [0,∞), and there exists at least one weak solution u(x,t) ∈ L∞([0,D]×R+;R) for the transport PDE part, yielding a solution X(t) ∈ AC(R+;R^n).

“Thus, a Filippov solution P ( t ) exists (under the local Lipschitzness assumptions on f and κ ) by, e.g., [14], which is absolutely continuous on [0 , ∞ ) .”

• cites [14] [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides , Springer Science & Business Media, 1988. - sourceNot checked — no full text of the source
• cites [26] [26] T. Liard, I. Balogoun, S. Marx, and F. Plestan, 'Boundary sliding mode control of a system of linear hyperbolic equations: A Lyapunov approach,' Automatica , vol. 135, Art. no. 109964, 2022. - sourceNot checked — no full text of the source

May combine two claims.

The control design in the input quantization case is different from the state quantization case: both the structure of the controller and the parameters of the zoom variable are different.

“The control design in the input quantization case is different than in the state quantization case. In particular, both the structure of the controller and the parameters of the zoom variable are different.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

System (2), (64) is in the strict-feedforward form, so the quantized predictor-feedback laws are given explicitly via (67) and (68).

“Since system (2), (64) is in the strict-feedforward form our quantized, predictor-feedback laws are given explicitly via (67) and (68).”

Probably not a substantial claim: may not pass S4 (based).

Such ˆα ∈ K∞ exists under Assumption 2.

“5 Such ˆ α ∈ K ∞ exists under Assumption 2.”

May combine two claims.

Typically, a sufficiently large M would guarantee condition (25), as even if M2 may depend on M via the term μ0 M1hat, μ0 M1hat can typically be upper bounded independently of M for large M, given that μ0 is chosen inversely proportional to M.

“Typically, a sufficiently large M would guarantee (25), as even if M 2 may potentially depend on M (via term µ 0 ˆ M 1 ) , it would typically be the case that µ 0 ˆ M 1 can be upper bounded independently of M for large M , given the choice of µ 0 being inversely proportional to M .”

May combine two claims.

In the input quantization case, the locally Lipschitz assumption on the quantizer is not required because the function subject to quantization can be treated as a finite-dimensional object (the exact predictor state), and well-posedness can be studied by combining results for ODE systems with [26].

“Nevertheless, in the input quantization case such an assumption is not required as the function that is subject to quantization can be treated as a finite-dimensional object (in fact, it is the exact predictor state), and thus, well-posedness can be studied combining results for ODE systems with, e.g., [26].”

• cites [26] [26] T. Liard, I. Balogoun, S. Marx, and F. Plestan, 'Boundary sliding mode control of a system of linear hyperbolic equations: A Lyapunov approach,' Automatica , vol. 135, Art. no. 109964, 2022. - sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The value of μ0 is analogous to the resulting overshoot of the closed-loop response; the ω variable is essentially an indicator of how fast the decrease of μ, and thus of the state, is; and the smaller the switching instants Ti, the faster the decrease of μ, and thus the faster the convergence of the state.

“In general, the three main components of the zoom mechanism have the following effects on closed-loop performance. The value of µ 0 is analogous to the resulting overshoot of the closed-loop response, the ω variables is, essentially, an indicator of how fast the decrease of µ , and thus, of the state, is, while the smaller the switching instants T i the faster the decrease of µ , and thus, the”

May combine two claims.

The result obtained is semiglobal asymptotic stability, because assumption (26) on the initial condition is required to hold for any given E0.

“The result we obtain is semiglobal asymptotic stability, as the assumption (26) on the initial condition is required to hold for any given E 0 .”

Probably not a substantial claim: may not pass S4 (based).

The locally Lipschitz assumption on the quantizer cannot be removed in general.

“Although such an assumption cannot be removed in general”

• AssumptionThe authors state this as an assumption: The (almost) quantizer is assumed to be a locally Lipschitz function, which guarantees well-posedness of the closed-loop system.

May combine two claims.

Condition (25) is a restriction on the quantizer's parameters M and ∆, and it can be satisfied, for example, for sufficiently large M and/or sufficiently small ∆.

“We note that condition (25) is a restriction on the quantizer's parameters M and ∆ , and it can be satisfied, for example, for sufficiently large M and/or sufficiently small ∆ .”

May combine two claims.

The core ingredients of the design approach are a quantized predictor-feedback law and a switching mechanism for the adjustable variable of the quantizer; the former achieves delay compensation through its predictor structure (as established in Theorems 1 and 2, which do not restrict the delay value), and the latter achieves compensation of the quantization effect by adjusting online the range/

“The core ingredients of our design approach are a) a quantized predictor-feedback law and b) a switching mechanism for the adjustable variable of the quantizer. The former component achieves delay compensation, as it exhibits a predictor structure (and as it is established in Theorems 1 and 2, which do not impose a restriction on the delay value). The latter component achieves compensation of the”

Classified as probably not a substantial claim.

The stability proofs rely on constructive derivation of solutions estimates, via a combination of input-to-state stability (ISS) arguments and backstepping transformations.

“The stability proofs rely on constructive derivation of solutions estimates, via a combination of input-to-state stability (ISS) arguments and backstepping transformations.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Implementation of nonlinear predictor feedbacks is thoroughly addressed in [17], and its practical feasibility is supported by experimental implementations [1], [32], [33].

“Implementation of nonlinear predictor feedbacks is thoroughly addressed in [17], while its practical feasibility is supported by experimental implementations [1], [32], [33].”

• cites [17] [17] I. Karafyllis and M. Krstic, Predictor Feedback for Delay Systems: Implementations and Approximations, Springer, 2017. - sourceNot checked — no full text of the source
• cites [1] [1] A. Bertino, P. Naseradinmousavi, and M. Krsti´ c, 'Delay-Adaptive Control of a 7-DOF Robot Manipulator: Design and Experiments,' IEEE Transactions on Control Systems Technology , vol. 30, no. 6, pp. 2506-2521, 2022. - sourceNot checked — no full text of the source
• cites [32] [32] A. Samii, R. de Haan, and N. Bekiaris-Liberis, 'Experimental implementation and validation of predictor-based CACC for vehicular platoons with distinct actuation delays,' IEEE CDC , 2025. - sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing).

The solutions of the closed-loop system with input quantization are to be interpreted in the sense of Filippov.

“Since the quantizer is a piecewise constant function the solutions of the closed-loop system are to be interpreted in the sense of Filippov.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The design and stability analysis do not explicitly depend on the generalized solutions interpretation, as also noted in [27], [28] for ODE systems, on which the well-posedness proof essentially relies via the predictor state P.

“We note that our design and stability analysis do not explicitly depend on such generalized solutions interpretation, as also noted in [27], [28] for the case of ODE systems, on which our well-posedness proof, essentially, relies (via considering the predictor state P ).”

• cites [27] [27] D. Liberzon, 'Hybrid feedback stabilization of systems with quantized signals,' Automatica , vol. 39, pp. 1543-1554, 2003. - sourceNot checked — no full text of the source
• cites [28] [28] D. Liberzon, 'Quantization, time delays, and nonlinear stabilization,' IEEE Trans. on Autom. Control , vol. 51, no. 7, pp. 1190-1195, 2006. - sourceNot checked — no full text of the source