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REVIEW 4 major objections 4 minor 59 references

Provably-Stable Neural Network-Based Control of Nonlinear Systems

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper proves that a neural network trained to imitate a one-step-ahead predictive controller, which outputs both a control action and a quadratic Lyapunov matrix, stabilizes a nonlinear system with a tunable tracking-error bound…

desk verdict The paper's core stability claim collapses at the target point: the proposed optimization is infeasible at equilibria unless the linearization error lies in the input range, so the main theorem is false. read the letter →

arxiv 2502.00248 v1 pith:3I6IS4FG submitted 2025-02-01 math.OC cs.LGcs.SYeess.SY

Anran Li , John P. Swensen , Mehdi Hosseinzadeh This is my paper · ORCID
classification math.OCcs.LGcs.SYeess.SY MSC 93D3093C1093C5568T07
keywords neural-network-basedcontrolstabilityguaranteesLyapunovmatrixone-step-aheadpredictiveaffinenonlinearsystemstrackingerrorboundmodelboundedapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's aim is to give neural-network-based control of nonlinear systems a stability guarantee it has lacked: if a trained network mimics a one-step-ahead predictive controller with bounded approximation error, the closed-loop states stay bounded and converge to a tunable neighborhood of the desired equilibrium. The predictive controller solves, at each step, a small optimization that returns both the control input and a positive-definite Lyapunov matrix for the current operating point. The paper then proves that a feedforward network imitating that input–output map keeps the system stable, with the eventual tracking-error radius expressed explicitly in terms of the linearization error, the network's worst-case output deviations, and the single design parameter $\theta$. Numerical studies on an inverted pendulum and hardware experiments on a quadrotor drone are offered as evidence that the scheme works in practice. A sympathetic reader would summarize the contribution as: imitation with bounded error is enough to preserve stability, and the degradation is controlled by one scalar design parameter.

What carries the argument

The central object is the optimization problem (3), whose decision variables are the control input $u$ and a positive-definite Lyapunov matrix $P$, with the Lyapunov decrease constraint $V(x^{+},r,P)-V(x,r,P)\le-\theta\|x-\bar{x}_r\|$ enforced along the one-step linearized prediction $x^{+}=A_t x(t)+B_t u$. This constraint is what makes $\theta$ a tunable contraction rate and what later lets the network's output errors be absorbed. The proof's load-bearing identity is the cost comparison (10): since $(u^*(t+1),P^*(t+1))$ is optimal at time $t+1$, it costs no more than reusing the previous $P^*(t)$, which transfers the Lyapunov decrease from the exact controller to the imitating network. The bounded-imitation assumption bounds $\Delta u(t)$ and $\Delta P(t)$ by $\bar{\Delta}_u$ and $\bar{\Delta}_P$, turning the Lyapunov difference into inequality (27), where every error term is multiplied by factors that $\theta$ must dominate.

What would settle it

Take any stabilizable discrete-time nonlinear system with nonzero linearization error, simulate the exact one-step-ahead controller (3), and at each time $t$ test whether the previously optimal $P^*(t)$ satisfies constraints (3c)–(3d) when evaluated at $x(t+1)$. If an instance appears where $P^*(t)$ is infeasible at $t+1$, the cost-comparison step (10) and the Lyapunov-decrease conclusion of Theorem 2 no longer follow, so the neural-network bound in Theorem 3 would not be covered by this proof.

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Extended reading notes

Core claim

The central claim is Theorem 3: for the discrete-time affine nonlinear system $x(t+1)=f(x(t))+g(x(t))u(t)$, if a neural network returns control and Lyapunov-matrix outputs $u^*(t)+\Delta u(t)$ and $P^*(t)+\Delta P(t)$ with worst-case deviations $\bar{\Delta}_u$ and $\bar{\Delta}_P$, then for any $\theta$ larger than the threshold in (27) the tracking error $\|x(t)-\bar{x}_r\|$ remains bounded and satisfies $\limsup_{t\to\infty}\|x(t)-\bar{x}_r\|\le\vartheta$, with $\vartheta$ given by equation (28) and made arbitrarily small by increasing $\theta$. The companion Theorem 2 establishes the same property for the exact one-step-ahead predictive controller, with error radius $\sigma=3\sqrt{\bar{\lambda}_P}\delta/\theta$, where $\delta$ bounds the linearization error and $\bar{\lambda}_P$ bounds the largest eigenvalue of the learned Lyapunov matrices. The mechanism is that each time step produces a fresh quadratic Lyapunov function tailored to the current operating point, so the stabilizing decrease condition (3d) holds along the linearized prediction; the network's imitation errors appear only as additive terms that the design parameter $\theta$ is chosen to dominate.

Load-bearing premise

The proof assumes that the Lyapunov matrix $P^*(t)$ that was optimal at time $t$ is still a feasible candidate at time $t+1$; the paper's feasibility argument only invokes stabilizability of the linearized pair, which does not by itself guarantee a positive-definite $P$ that continues to satisfy the decrease constraint (3d).

Editorial extensions

If this is right

  • A neural network trained offline by solving (3) on a grid can replace the online solver in the loop while keeping bounded tracking error, provided the network's worst-case output deviations satisfy the threshold in (27).
  • Increasing the single scalar $\theta$ shrinks the guaranteed asymptotic ball radius $\vartheta$ toward zero; the paper notes that large $\theta$ can make the optimization numerically ill-conditioned.
  • For linear systems the linearization error $\delta$ is zero, so the exact one-step controller converges asymptotically to the equilibrium, and the scheme reduces to a one-step finite-horizon linear-quadratic regulator.
  • Because the stability certificate depends only on bounded output deviations, the same guarantee would hold for any imitation of the predictive controller, not only the specific feedforward network used in the experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit bound (28) suggests a design loop the paper does not spell out: choose $\theta$ from the desired tracking radius, then train the network until its worst-case deviations $(\bar{\Delta}_u,\bar{\Delta}_P)$ satisfy the threshold in (27).
  • Because the theorem uses only bounded output deviations, the same guarantee should transfer to other cheap imitations of the predictive controller, such as quantized networks or lookup-table policies.
  • A natural next step is to check numerically whether $P^*(t)$ remains feasible at $t+1$ across a range of nonlinear systems, which would turn the assumed cost comparison into a verifiable certificate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a one-step-ahead predictive control scheme with a jointly optimized control input and quadratic Lyapunov matrix, then trains a neural network to imitate the resulting policy and claims provable stability and tracking bounds for the NN-controlled closed loop. The main theoretical results are Theorem 1 (recursive feasibility), Theorem 2 (stability/convergence of the predictive controller), and Theorem 3 (boundedness of the tracking error with the NN in the loop, with an explicit bound on the ultimate ball). The paper also includes simulation studies on an inverted pendulum and experiments on a Parrot Bebop 2 drone.

Significance. The problem addressed is important: NN-based control with rigorous stability guarantees is an active and practically relevant area. The paper has several strengths: the authors provide a repository with code and data, include both simulation and experimental validation, and the proposed idea of quantifying the degradation caused by NN approximation error with a single design parameter is attractive. However, the central theoretical claims are not established. The recursive-feasibility theorem is false as stated, the proof of Theorem 2 relies on unproven eigenvalue bounds and an unjustified candidate-feasibility step, and Theorem 3 uses norm-like quantities for possibly indefinite matrices. Because the main contribution is the stability proof, these gaps are load-bearing and the current version cannot be recommended for publication.

major comments (4)
  1. [Section 3.2, Theorem 1] The proof of recursive feasibility claims that stabilizability of (A_t, B_t) (Assumption 1) is sufficient for feasibility of (3). This is false. At x(t) = xbar_r, constraint (3d) becomes V(x^+, r, P) - V(x(t), r, P) <= 0, and since V(x(t), r, P)=0 and V >= 0, it forces x^+ = xbar_r. Constraint (3b) then requires xbar_r = A_t xbar_r + B_t u, which is solvable only if f(xbar_r) - A_t xbar_r lies in Im(B_t); neither Assumption 1 nor Assumption 2 implies this. A concrete counterexample satisfying Assumptions 1-3 is n=2, p=1, f(x) = (x1 + x1^2 - 1 + c x2, x2), g(x) = (0,1)^T, xbar_r = (1,0), ubar_r = 0, c != 0. The linearized pair at xbar_r is stabilizable, but at x = xbar_r the predicted state is (3,u)^T, which never equals (1,0), so no P > 0 satisfies (3d). Thus Theorem 1 is not proven, and the recursive feasibility on which Theorems 2 and 3 depend is not available.
  2. [Section 3.2, Theorem 2 proof, Eq. (10)] The inequality J(u^*(t+1), P^*(t+1)|x(t+1), r) <= J(u^*(t+1), P^*(t)|x(t+1), r) in Eq. (10) is only valid if (u^*(t+1), P^*(t)) is feasible for (3) at time t+1. The paper does not show that the optimal Lyapunov matrix P^*(t) from time t satisfies constraints (3c)-(3d) at time t+1; Theorem 1 was supposed to establish this, but its proof is invalid as shown above. Furthermore, the proof introduces lambda_P = inf_t lambda_min(P^*(t)) and lambda_bar_P = sup_t lambda_max(P^*(t)) and assumes 0 < lambda_P <= lambda_bar_P < infinity, yet nothing in (3) enforces uniform eigenvalue bounds on P^*(t). Since P is a free decision variable with only P > 0 and the decrease constraint (3d), P^*(t) could have eigenvalues tending to zero or to infinity over time. Consequently, the constants sigma in Theorem 2 and the bound in Theorem 3 are not well defined.
  3. [Section 5.1, Theorem 3 proof, Eqs. (17)-(27)] The proof repeatedly applies norm inequalities to || . ||_{P^*(t)+Delta P(t)} even though the paper explicitly states that P_hat(t) = P^*(t) + Delta P(t) is not necessarily positive definite. The quantity ||z||_Q = sqrt(|z^T Q z|) is not a norm when Q is indefinite, and the triangle-type inequalities in footnote 6 do not hold in general. For example, the transition from (18) to (19) and the upper bound of the fourth term in (21)-(23) are not justified for indefinite Q. This invalidates the chain of inequalities leading to (27) and the explicit formula for vartheta in (28). Since Theorem 3 is the paper's main NN-stability result, this is a fundamental gap.
  4. [Sections 3.1 and 5.1, Remarks 4 and 7] The paper claims that sigma and vartheta can be made arbitrarily small by increasing the design parameter theta. However, theta appears in the constraint (3d) itself; increasing theta requires a larger one-step decrease of V and can render problem (3) infeasible. In the counterexample in the first major comment, there is no value of theta for which (3) is feasible at xbar_r. Thus the statement 'for sufficiently large theta' in Theorem 3 is not a license to choose theta freely, and the claim of an arbitrarily small tracking ball is unsupported.
minor comments (4)
  1. [Throughout] There are several typographical errors: 'Laypunov' should be 'Lyapunov', 'optimizaiton' in Remark 4 should be 'optimization', 'Y ALMIP' should be 'YALMIP', 'Pytorch' should be 'PyTorch', and 'cade' in the Data Availability statement should be 'code'.
  2. [Section 5.2, Eq. (29)] The definition of the region of attraction uses the predicted state x_hat(k|x,r) without specifying whether the prediction is generated by the linearized model in (3b) or by the original nonlinear dynamics (1), and which control law is used in the prediction. This needs to be clarified for the set Phi(r) to be well-defined.
  3. [Section 6.2] The reported mean computing time of 0.565 seconds for the NN-based scheme is larger than the sampling time of 0.1 seconds used in the discretization (32); the claim that the NN scheme is suitable for real-time implementation should be reconciled with these numbers.
  4. [Remark 1] The claim in Remark 1 that the constraint (3d) imposes exponential stability when sqrt(lambda_min(P)) > theta is stated without proof or reference to a precise stability definition; a short derivation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: stability bounds are analytic consequences of the Lyapunov-decrease constraint, not reductions to fitted inputs or self-citations; the identified proof gaps are soundness failures rather than circular reasoning.

full rationale

No circular derivation is present. The optimization problem (3) explicitly enforces the Lyapunov decrease constraint (3d), and the subsequent Theorems 2 and 3 analyze the consequences of that constraint along the true nonlinear dynamics and under neural-network approximation error, rather than restating the constraint as a conclusion. The derived bounds, sigma = 3 sqrt(lambda_bar_P) delta / theta and vartheta in (28), depend on the model linearization error delta, the NN approximation errors Delta_u and Delta_P, Lipschitz constants, and the user-selected parameter theta; no term is obtained by fitting data, and theta is not calibrated to realize a pre-specified bound. The self-citations in the paper are routine and not load-bearing. The serious weaknesses are soundness gaps, not circularity: Theorem 1 infers recursive feasibility of (3) solely from stabilizability without proving the existence of P satisfying (3d); inequality (10) requires P*(t) to be feasible at time t+1 without establishing that property; and at x = xbar_r, constraint (3d) forces x+ = xbar_r while (3b) requires f(xbar_r) - A xbar_r in Im(B(xbar_r)), which Assumption 2 does not guarantee. These are genuine correctness concerns, but in none of them is the claimed conclusion used as an input to its own proof, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The stability guarantee rests on standard assumptions about the system, plus several unverified assumptions about the optimizer and the trained network. Most importantly, the proof assumes recursive feasibility and bounded eigenvalues of P*, and the NN error bounds are assumed rather than certified. The user-selected parameters theta, Qx, Qu and the NN architecture are inputs that shape the data and the bound, but they are not fit to the claimed result.

free parameters (3)
  • theta = 0.01, 0.001, 0.0001 in pendulum; 1 in drone
    User-selected trade-off between tracking bound and numerical conditioning; the theory requires it to be large, while Remark 4 warns large theta causes ill-conditioning.
  • Qx and Qu weighting matrices = Qx = 2I2, Qu = 0.1 (pendulum); Qx = 20I2, Qu = 0.1 (drone)
    User-selected weights in the optimization (3); they affect the data and the imitation target.
  • NN architecture hyperparameters = 6 hidden layers with 8, 32, 64, 64, 32, 16 neurons; Adam, lr=0.001, 10000 epochs
    Chosen by the growing method; not part of the theory, but shapes the approximation error that the bound depends on.
assumptions (6)
  • domain assumption Assumption 1: the linearized pair (df/dx, g(x)) is stabilizable at every x in X.
    Needed so that a stabilizing control exists for the linearized model underlying (3).
  • domain assumption Assumption 2: the linearization error ||f(x)-A x|| <= delta for all x in X with finite known delta.
    delta enters the tracking bounds sigma and vartheta; the proof requires delta finite and known.
  • domain assumption Assumption 3: f and g are Lipschitz on X with constants mu_f and mu_g.
    Lipschitz continuity of g is used to bound the effect of NN input error in equation (23).
  • ad hoc to paper Bounded NN approximation error: sup_t ||Delta u(t)|| = Delta_bar_u and sup_t ||Delta P(t)|| = Delta_bar_P are finite.
    Stated as 'reasonable' in Section 5; no verification method is provided, and it is essential for Theorem 3.
  • ad hoc to paper Recursive feasibility and global optimality of the optimization problem (3).
    Theorem 1 does not prove recursive feasibility, and YALMIP is not guaranteed to find a global optimum of the non-convex problem; the proofs use optimality comparisons that require the global optimum.
  • ad hoc to paper Bounded spectrum of P*(t): lambda_P > 0 and lambda_bar_P < infinity.
    The bounds in Theorems 2 and 3 are expressed in terms of these quantities, but nothing in (3) constrains the eigenvalues of P.

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Pith. "Pith review of Provably-Stable Neural Network-Based Control of Nonlinear Systems." pith.science (2026). https://pith.science/paper/3I6IS4FG

@misc{pith2026250200248,
  author       = {Pith},
  title        = {Pith review of: Provably-Stable Neural Network-Based Control of Nonlinear Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3I6IS4FG}},
  note         = {Machine review of arXiv:2502.00248}
}
read the original abstract

In recent years, Neural Networks (NNs) have been employed to control nonlinear systems due to their potential capability in dealing with situations that might be difficult for conventional nonlinear control schemes. However, to the best of our knowledge, the current literature on NN-based control lacks theoretical guarantees for stability and tracking performance. This precludes the application of NN-based control schemes to systems where stringent stability and performance guarantees are required. To address this gap, this paper proposes a systematic and comprehensive methodology to design provably-stable NN-based control schemes for affine nonlinear systems. Rigorous analysis is provided to show that the proposed approach guarantees stability of the closed-loop system with the NN in the loop. Also, it is shown that the resulting NN-based control scheme ensures that system states asymptotically converge to a neighborhood around the desired equilibrium point, with a tunable proximity threshold. The proposed methodology is validated and evaluated via simulation studies on an inverted pendulum and experimental studies on a Parrot Bebop 2 drone.

Figures

Figures reproduced from arXiv: 2502.00248 by the authors.

Figure 1
Figure 1. General structure of proposed methodology. control scheme for nonlinear systems and analytically proving its theoretical properties; and ii) developing a NN-based control scheme for nonlinear systems, formally proving its stability and convergence properties, and evaluating the effectiveness of the proposed control schemes via simulation and experimental studies. The main features of the propose NN-based control sch… view at source ↗
Figure 2
Figure 2. The NN trained to approximate the solution of (3), with s hidden layers and li neurons in the ith hidden layer. 5.1. Theoretical Analysis The following theorem shows that if the trained NN is utilized in the loop to control system (1), the system is stable and the tracking error remains bounded. See Figure1 for the general structure of the proposed NN-based control scheme. Theorem 3. Consider system (1), and suppose… view at source ↗
Figure 3
Figure 3. Geometric illustration of the falsification procedure for determining the RoA Φ(r). where α represents angle of the inverted pendulum, τ is the torque generated from a motor rotating the pendulum, and m = 1 [kg] and L = 1 [m] are the pendulum mass and the distance from the center of mass, respectively. We assume that X = [−5, 5]×[−5, 5] and R = [−1, 1]. The state-space representation for this system is: x˙1 = x2, (3… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Inverted pendulum system [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: The obtained RoA with θ = 0.01 and r = 0; the blue and yellow represent X and Φ(0), respectively. 6.2. Performance Analysis To evaluate the effectiveness of the proposed one-step-ahead predictive control scheme and the NN-based control scheme, we consider three cases b…
Figure 6
Figure 6. Figure 6: Time profile of x(t) with θ = 0.0001. Note that NN-based control scheme is faster than the one-step-ahead control scheme, and thus is appropriate for real-time applications. For instance, when θ = 0.01, the mean computing time for the NN-based control scheme is 0.565 s…
Figure 7
Figure 7. Figure 7: Time profile of state vector x(t) and control input u(t). Also, Figure7 reveals that the iterative LQR technique provides a better solution in compari￾son with the proposed NN-based control scheme; note that this observation is understandable, as the linearization erro…
Figure 8
Figure 8. Figure 8: Phase portrait graph [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Time profile of ∆V(t) [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Time profile of the drone’s position [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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Works this paper leans on

59 extracted references · 53 canonical work pages

  1. [1]

    H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ, USA: Prentice Hall, 2002. 19

    work page 2002

  2. [2]

    Isidori, Nonlinear Control Systems

    A. Isidori, Nonlinear Control Systems. London: Springer Verlag, 1995

    work page 1995

  3. [3]

    Ogata, Modern Control Engineering

    K. Ogata, Modern Control Engineering. Prentice Hall, 2010

    work page 2010

  4. [4]

    Chen, Linear System Theory and Design

    C.-T. Chen, Linear System Theory and Design. Oxford University Press, 1998

    work page 1998

  5. [5]

    Iterative linear quadratic regulator design for nonlinear biological movement systems,

    W. Li and E. Todorov, “Iterative linear quadratic regulator design for nonlinear biological movement systems,” in Proc. 1st International Conference on Informatics in Control, Automation and Robotics, 2004, pp. 222–229

    work page 2004

  6. [6]

    A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems,

    E. Todorov and W. Li, “A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems,” in Proc. American Control Conference, Portland, OR, USA, Jun. 8–10, 2005, pp. 300–306

    work page 2005

  7. [7]

    Design of a linear quadratic regulator for nonlinear systems modeled via norm-bounded linear differential inclusions,

    C. R. Rodrigues, R. Kuiava, and R. A. Ramos, “Design of a linear quadratic regulator for nonlinear systems modeled via norm-bounded linear differential inclusions,” IFAC Proceedings Volumes, vol. 44, no. 1, pp. 7352–7357, Jan. 2011

    work page 2011

  8. [8]

    Optimal control of nonlinear inverted pendulum system using PID con- troller and LQR: Performance analysis without and with disturbance input,

    L. B. Prasad, B. Tyagi, and H. O. Gupta, “Optimal control of nonlinear inverted pendulum system using PID con- troller and LQR: Performance analysis without and with disturbance input,” International Journal of Automation and Computing, vol. 11, pp. 661–670, 2014

    work page 2014

Show all 59 references
  1. [9]

    Robust adaptive LQR control of nonlinear system application to 3-Dof flight control system,

    R. I. Boby, H. Mansor, T. S. Gunawan, and S. Khan, “Robust adaptive LQR control of nonlinear system application to 3-Dof flight control system,” in Proc. IEEE International Conference on Smart Instrumentation, Measurement and Applications, Kuala Lumpur, Malaysia, Nov. 25, 2014

    2014

  2. [10]

    Finite-time stabilization of nonlinear time delay systems using LQR based sliding mode control,

    K. Mathiyalagan and G. Sangeetha, “Finite-time stabilization of nonlinear time delay systems using LQR based sliding mode control,” Journal of the Franklin Institute, vol. 356, no. 7, pp. 3948–3964, May 2019

    2019

  3. [11]

    Constrained iterative LQR for on-road autonomous driving motion planning,

    J. Chen, W. Zhan, and M. Tomizuka, “Constrained iterative LQR for on-road autonomous driving motion planning,” in Proc. IEEE 20th International Conference on Intelligent Transportation Systems, Yokohama, Japan, Oct. 16–19, 2017, pp. 1–7

    2017

  4. [12]

    Sastry, Nonlinear Systems: Analysis, Stability, and Control

    S. Sastry, Nonlinear Systems: Analysis, Stability, and Control. Springer New York, NY , 1999

    1999

  5. [13]

    Nonlinear model predictive control: A tutorial and survey,

    J. B. Rawlings, E. S. Meadows, and K. R. Muske, “Nonlinear model predictive control: A tutorial and survey,” IFAC Proceedings Volumes, vol. 27, no. 2, pp. 185–197, May 1994

    1994

  6. [14]

    Vidyasagar, Nonlinear Systems Analysis

    M. Vidyasagar, Nonlinear Systems Analysis. Prentice Hall, 1993

    1993

  7. [15]

    K. J. Hunt, G. R. Irwin, and K. Warwick, Neural Network Engineering in Dynamic Control Systems . Springer London, 1995

    1995

  8. [16]

    Adaptive robust model predictive control with matched and unmatched uncertainty,

    R. Sinha, J. Harrison, S. M. Richards, and M. Pavone, “Adaptive robust model predictive control with matched and unmatched uncertainty,” inProc. American Control Conference, Atlanta, GA, USA, Jun. 8–10, 2021, pp. 906–913

    2021

  9. [17]

    A robust neural network approximation-based prescribed performance output-feedback controller for autonomous underwater vehicles with actuators saturation,

    O. Elhaki and K. Shojaei, “A robust neural network approximation-based prescribed performance output-feedback controller for autonomous underwater vehicles with actuators saturation,” Engineering Applications of Artificial Intelligence, vol. 88, p. 103382, 2020

    2020

  10. [18]

    Neural-networks control for hover to high-speed-level-flight transition of ducted fan uav with provable stability,

    Z. Cheng, H. Pei, and S. Li, “Neural-networks control for hover to high-speed-level-flight transition of ducted fan uav with provable stability,”IEEE Access, vol. 8, pp. 100 135–100 151, 2020

    2020

  11. [19]

    Control of a tail-sitter VTOL UA V based on recurrent neural networks,

    J. Zhou, H. Xu, Z. Li, S. Shen, and F. Zhang, “Control of a tail-sitter VTOL UA V based on recurrent neural networks,” arXiv preprint arXiv:2104.02108, 2021

    2021 arXiv

  12. [20]

    Design, implementation, and evaluation of a neural-network-based quadcopter UA V system,

    F. Jiang, F. Pourpanah, and Q. Hao, “Design, implementation, and evaluation of a neural-network-based quadcopter UA V system,”IEEE Transactions on Industrial Electronics, vol. 67, no. 3, pp. 2076–2085, 2019

    2019

  13. [21]

    Punishment di- minishes the benefits of network reciprocity in social dilemma experiments,

    X. Li, M. Jusup, Z. Wang, H. Li, L. Shi, B. Podobnik, and H. Stanley, S. Havlin, and S. Boccaletti, “Punishment di- minishes the benefits of network reciprocity in social dilemma experiments,”Proceedings of the National Academy of Sciences, vol. 115, pp. 20–35, Feb. 2018

    2018

  14. [22]

    Lyapunov-stable neural-network control,

    H. Dai, B. Landry, L. Yang, M. Pavone, and R. Tedrake, “Lyapunov-stable neural-network control,”arXiv preprint arXiv:2109.14152, 2021

    2021 arXiv

  15. [23]

    Augmented neural Lyapunov control,

    D. Grande, A. Peruffo, E. Anderlini, and G. Salavasidis, “Augmented neural Lyapunov control,” IEEE Access, vol. 11, pp. 67 979–67 986, 2023

    2023

  16. [24]

    A LQR-based neural-network controller for fast stabilizing rotary inverted pendulum,

    H. V . Nghi, D. Phuoc Nhien, N. T. Minh Nguyet, N. Tu Duc, N. P. Luu, P. Son Thanh, L. T. Hong Lam, and D. Xuan Ba, “A LQR-based neural-network controller for fast stabilizing rotary inverted pendulum,” in Porc. International Conference on System Science and Engineering, Ho Ch...

    2021

  17. [25]

    A new neural network-based optimal mixed H2/H∞ control for a modified unmanned aerial vehicle subject to control input constraints,

    F. W. Alsaade, H. Jahanshahi, Q. Yao,M. S. Al-zahrani, and A. S. Alzahrani, “A new neural network-based optimal mixed H2/H∞ control for a modified unmanned aerial vehicle subject to control input constraints,” Advances in Space Research, vol. 71, no. 9, pp. 3631–3643, 2023

    2023

  18. [26]

    Lyapunov-based continuous-time nonlinear control using deep neural network applied to underactuated systems,

    R. C. Rego and F. M. U. de Araujo, “Lyapunov-based continuous-time nonlinear control using deep neural network applied to underactuated systems,” Engineering Applications of Artificial Intelligence, vol. 107, p. 104519, 2022

    2022

  19. [27]

    Esfandiari, F

    K. Esfandiari, F. Abdollahi, and H. A. Talebi, Neural Network-Based Adaptive Control of Uncertain Nonlinear Systems. Springer, Cham, 2021

    2021

  20. [28]

    es-DNLC: A deep neural network control with exponentially stabilizing control lyapunov functions for attitude stabilization of PA V,

    M. Jang, J. Hyun, T. Kwag, C. Gwak, C. Jeong, T. A. Nguyen, and J.-W. Lee, “es-DNLC: A deep neural network control with exponentially stabilizing control lyapunov functions for attitude stabilization of PA V,” in Proc. 22nd 20 International Conference on Control, Automation an...

    2022

  21. [29]

    UAMDynCon-DT: A data-driven dynamics and robust control framework for UAM vehicle digitalization using deep learning,

    M. Jang, J. Hyun, T. Kwag, C. Gwak, T. A. Nguyen, and J.-W. Lee, “UAMDynCon-DT: A data-driven dynamics and robust control framework for UAM vehicle digitalization using deep learning,” in Proc. International Confer- ence on Mechatronics, Control and Robotics, 2023, pp. 81–85

    2023

  22. [30]

    Enforcing robust control guarantees within neural network policies,

    P. L. Donti, M. Roderick, M. Fazlyab, and J. Z. Kolter, “Enforcing robust control guarantees within neural network policies,” arXiv preprint arXiv:2011.08105, 2020

    2011 arXiv

  23. [31]

    Development of a neural network-based adaptive nonlinear dynamic inversion controller for a tilt-wing VTOL aircraft,

    J. Autenrieb, H.-S. Shin, and M. Bacic, “Development of a neural network-based adaptive nonlinear dynamic inversion controller for a tilt-wing VTOL aircraft,” inProc. Workshop on Research, Education and Development of Unmanned Aerial Systems, 2019, pp. 44–52

    2019

  24. [32]

    Learning control lyapunov functions from counterexamples and demonstrations,

    H. Ravanbakhsh and S. Sankaranarayanan, “Learning control lyapunov functions from counterexamples and demonstrations,” Autonomous Robots, vol. 43, pp. 275–307, 2019

    2019

  25. [33]

    E. F. Camacho, C. Bordons, E. F. Camacho, and C. Bordons, Model predictive controllers. Springer, 2007

    2007

  26. [34]

    J. B. Rawlings, D. Q. Mayne, and M. Diehl, Model predictive control: theory, computation, and design. Nob Hill Publishing Madison, WI, 2017, vol. 2

    2017

  27. [35]

    Grune and J

    L. Grune and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms. Springer-Verlag London, 2011

    2011

  28. [36]

    Allgower and A

    F. Allgower and A. Zheng, Nonlinear Model Predictive Control. Springer Science & Business Media, 2000

    2000

  29. [37]

    Robust to early termination model predictive control,

    M. Hosseinzadeh, B. Sinopoli, I. Kolmanovsky, and S. Baruah, “Robust to early termination model predictive control,” IEEE Transactions on Automatic Control, 2023, DOI: 10.1109/TAC.2023.3308817

    2023

  30. [38]

    A control-theoretic approach for dynamic adaptive video streaming over HTTP,

    X. Yin, A. Jindal, V . Sekar, and B. Sinopoli, “A control-theoretic approach for dynamic adaptive video streaming over HTTP,” inProc. ACM Conf. Special Interest Group on Data Communication, London, United Kingdom, Aug. 17-21, 2015, pp. 325–338

    2015

  31. [39]

    CANE: A cascade-control approach for network-assisted video QoE management,

    M. Hosseinzadeh, K. Shankar, M. A. abd Jay Ramachandran, S. Adams, V . Sekar, and B. Sinopoli, “CANE: A cascade-control approach for network-assisted video QoE management,” IEEE Transactions on Control Systems Technology, vol. 31, no. 6, pp. 2543–2554, Nov. 2023

    2023

  32. [40]

    A stable one-step-ahead predictive control of non-linear systems,

    C. Kambhampati, J. D. Mason, and K. Warwick, “A stable one-step-ahead predictive control of non-linear systems,” Automaticas, vol. 36, no. 4, pp. 485–495, Apr. 2000

    2000

  33. [41]

    On the exponential stability of discrete-time systems with applications in observer design,

    V . C. Aitken and H. M. Schwartz, “On the exponential stability of discrete-time systems with applications in observer design,” IEEE Transactions on Automatic Control, vol. 39, no. 9, pp. 1959–1962, Sep. 1994

    1959

  34. [42]

    T. L. Fine, Feedforward Neural Network Methodology. Springer New York, NY , 1999

    1999

  35. [43]

    Goodfellow, Y

    I. Goodfellow, Y . Bengio, and A. Courville, Deep Learning. MIT Press, 2016

    2016

  36. [44]

    GradMax: Growing neural net- works using gradient information,

    U. Evci, B. van Merrienboer, T. Unterthiner, F. Pedregosa, and M. Vladymyrov, “GradMax: Growing neural net- works using gradient information,” in Proc. 10th International Conference on Learning Representations , Apr. 25–29, 2022

    2022

  37. [45]

    Firefly neural architecture descent: a general approach for growing neural networks,

    L. Wu, B. Liu, P. Stone, and Q. Liu, “Firefly neural architecture descent: a general approach for growing neural networks,” in Proceedings of the Advances in Neural Information Processing Systems 33, Dec. 6–12, 2020

    2020

  38. [46]

    Convex neural networks,

    Y . Bengio, N. Roux, P. Vincent, O. Delalleau, and P. Marcotte, “Convex neural networks,” in Proc. Advances in Neural Information Processing Systems, Vancouver, BC, Canada, 2005

    2005

  39. [47]

    Dropout: a simple way to prevent neural networks from overfitting,

    N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,”The journal of machine learning research, vol. 15, no. 56, pp. 1929–1958, 2014

    1929

  40. [48]

    A study on dropout techniques to reduce overfitting in deep neural networks,

    H.-i. Lim, “A study on dropout techniques to reduce overfitting in deep neural networks,” inProc. 14th International Conference on Multimedia and Ubiquitous Engineering, Jeju, South Korea, Aug. 17–19, 2020, pp. 133–139

    2020

  41. [49]

    Weight dropout for preventing neural networks from overfit- ting,

    K. Sanjar, A. Rehman, A. Paul, and K. JeongHong, “Weight dropout for preventing neural networks from overfit- ting,” in Proc. 8th International conference on orange technology, Daegu, Korea (South), Dec. 18–21, 2020

    2020

  42. [50]

    Towards dropout training for convolutional neural networks,

    H. Wu and X. Gu, “Towards dropout training for convolutional neural networks,” Neural Networks, vol. 71, pp. 1–10, Nov. 2015

    2015

  43. [51]

    Super convergence cosine annealing with warm-up learning rate,

    Z. Liu, “Super convergence cosine annealing with warm-up learning rate,” in Proc. 2nd International Conference on Artificial Intelligence, Big Data and Algorithms, Nanjing, China, Jun. 17–19, 2022, pp. 768–774

    2022

  44. [52]

    Navigating local minima in quantized spiking neural networks,

    J. K. Eshraghian, C. Lammie, M. R. Azghadi, and W. D. Lu, “Navigating local minima in quantized spiking neural networks,” in Proc. IEEE 4th International Conference on Artificial Intelligence Circuits and Systems , Incheon, Korea, Republic of, Jun. 13–15, 2022, pp. 352–355

    2022

  45. [53]

    Computing the n-dimensional delaunay tessellation with application to voronoi polytopes,

    D. F. Watson, “Computing the n-dimensional delaunay tessellation with application to voronoi polytopes,” The Computer Journal, vol. 24, no. 2, pp. 167–172, 1981

    1981

  46. [54]

    Properties of n-dimensional triangulations,

    C. L. Lawson, “Properties of n-dimensional triangulations,” Computer Aided Geometric Design, vol. 3, no. 4, pp. 231–246, Dec. 1986

    1986

  47. [55]

    The inverted pendulum benchmark in nonlinear control theory: a survey,

    O. Boubaker, “The inverted pendulum benchmark in nonlinear control theory: a survey,” International Journal of Advanced Robotic Systems, vol. 10, no. 5, 2013

    2013

  48. [56]

    Y ALMIP: a toolbox for modeling and optimization in MATLAB,

    J. Lofberg, “Y ALMIP: a toolbox for modeling and optimization in MATLAB,” in Proc. IEEE International Con- ference on Robotics and Automation, Taipei, Taiwan, Sep. 2–4, 2004, pp. 284–289. 21

    2004

  49. [57]

    https://pytorch.org, [Accessed February 4, 2025]

    2025

  50. [58]

    Adam: A method for stochastic optimization,

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980v9, 2017

    2017 arXiv

  51. [59]

    Closed-loop model identification and MPC-based navigation of quadcopters: A case study of parrot bebop 2,

    M. Amiri and M. Hosseinzadeh, “Closed-loop model identification and MPC-based navigation of quadcopters: A case study of parrot bebop 2,” arXiv:2404.07267, 2024. [Online]. Available: https://arxiv.org/abs/2404.07267 22

    2024 arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.