Neural Network-Aided Sampled-Data Control for a Class of Nonlinear Systems With Unknown Dynamics

May combine two claims.

For the Chua's circuit system, the LMIs in Theorem 1 are feasible for controller gains when the sampling period h is in the range (0, 1.436], guaranteeing exponential stabilization of the closed-loop system via sampled-data controller (9).

“For system (1), the LMIs in Theorem 1 are feasible for controller gains when h is in the range (0, 1.436], guaranteeing exponential stabilization of the closed-loop system via sampled-data controller (9), and the sampling period is h = 1.436s.”

May combine two claims.

In the numerical simulation, at t = 100 s the system states converge with high accuracy: |x1(t)| = 1.34×10^-17 V, |x2(t)| = 1.38×10^-17 V, and |x3(t)| = 2.60×10^-17 A.

“In particular, at t = 100s, |x 1 (t)| = 1.34 × 10 -17 V, |x 2 (t)| = 1.38 × 10 -17 V, and |x 3 (t)| = 2.60 × 10 -17 A, which illustrate the high convergence accuracy of system states.”

May combine two claims. The quote only partly states this claim.

The observer estimation error ξ and the neural network weight estimation error θ̃_f are uniformly ultimately bounded.

“We can infer that θ f uniformly ultimately bounded, meaning θ f ≤ , where depend on the initial value θ f (0).”

The quote only partly states this claim.

An L-K functional is constructed for the closed-loop system, leading to a sufficient stability condition in the form of an LMI.

“Third, an L-K functional is constructed, leading to a stability condition in the form of an LMI.”

The quote only partly states this claim.

Under Assumption 1, applying the ATASDC law (11) to system (1), if there exist positive-definite matrices S and W_0,...,W_{n-1}, R_1,...,R_{n-1} such that the LMI Ξ<0 holds, then for any positive constant δ>0 and any initial condition (x(0), θ_f(0)), the 2-norm of the state estimation error x̃(t) converges to the set Ω_s at an exponential rate θ*.

“By applying the ATASDC law (11) to system (1), it concludes that if there exist positive-definite matrices S ∈ R n×n and W 0 , W 1 , . . ., W n-1 , R 1 , R 2 , . . ., R n-1 ∈ R m×m such that the subsequent LMI holds:”

• Also stated in: § D. Stability Guarantee · plain

  “then for any positive constant δ > 0 and ∀(x(0), θ f (0)) ∈ R n × R, the 2-norm of x(t) converges to the set Ω s := {x ∈ R n | |x i | ≤ δ * } at an exponential rate denoted by the decay constant ϑ *”

• AssumptionThe authors state this as an assumption: The system (1) with controllable pair (A, B) and observable pair (C, A) possesses relative degree n ≥ 2.
• AssumptionThe authors state this as an assumption: f(y) is an unknown continuous nonlinear function defined over the compact set Ω_f.

May combine two claims.

The proposed method avoids requiring precise nonlinearity knowledge by approximating functions via RBFNNs using only sampled outputs, thereby reducing computational complexity.

“Unlike existing approaches [14], [25], the method avoids requiring precise nonlinearity knowledge by approximating functions via RBFNNs using only sampled outputs, thereby reducing computational complexity.”

• cites [14] sourceNot checked — no full text of the source
• cites [25] sourceNot checked — no full text of the source

Classified as probably not a substantial claim.

Leveraging the universal approximation property, the RBFNN approximates any continuous function f(y) over R^p as f(y) = θ*_f Ψ_f(y) + ε.

“Leveraging its universal approximation property, the RBFNN approximates any continuous function f (y) over R p as f (y) = θ * f Ψ f (y) + , ∀y ∈ R p .”