SidraBhatti/lnn-ismc-uav-control

Liquid Neural Network-Based Integral Sliding Mode Control of Multirotor UAV

A Liquid Neural Network (LNN)-based disturbance/uncertainty estimator integrated with an Integral Sliding Mode Controller (ISMC) for precise attitude and position tracking of a multirotor UAV under model uncertainty and external disturbance.

Fig. 2: Overall system — outer-loop (position) and inner-loop (attitude/altitude) controllers generate torque commands to the UAV; the LNN observer estimates states and disturbance from measured p, q, r and feeds corrected attitude/altitude estimates back into the inner loop.

Role & Attribution

Sidra Ghayour Bhatti — supervisory/co-advising role, alongside Qadeer Ahmed (OSU Dept. of Mechanical and Aerospace Engineering). Lead author and primary implementation: Zainab Akhtar (Mechatronics and Control Engineering, University of Engineering and Technology, Lahore), with Salman Ijaz (Control Systems Laboratory, University of Nottingham Ningbo, China) as co-supervisor/collaborator.

Motivation

UAVs have nonlinear, unstable dynamics sensitive to disturbances and model uncertainty. Classical observer-based approaches (Luenberger, Kalman filters, sliding-mode/high-gain observers) degrade under strong nonlinearities and rely on accurate mathematical models that are hard to derive for complex UAV dynamics. Prior neural-network estimators (including LSTM/GRU-based RNNs) improve robustness but face high computational cost, limited generalization, or gradient-related issues under fast-varying dynamics. This paper is the first application of Liquid Neural Networks — whose neurons behave as first-order dynamical systems governed by differential equations rather than static activation functions — to estimating uncertainty and disturbance in a nonlinear dynamical system.

Method

Quadrotor model. Standard 6-DOF attitude/position dynamics (roll, pitch, yaw angular rates; x/y/z position), with residual propeller speed Ω_r and combined uncertainty/disturbance term ζ_N(x,u) = Δg(x,u) + d(t) treated as unknown and estimated online — practically motivated by not wanting to load the UAV with extra sensors.

LNN architecture. A single-layer liquid (reservoir) network with 32 recurrent neurons. Unlike standard RNNs, each neuron's state evolves via τż = −z(t) + σ(W_in·x_in(t) + W_rec·z(t) + b), i.e. a first-order ODE with its own time constant τ, making the network inherently responsive to non-stationary inputs. The disturbance estimate is read out as ζ̂_N(x,u) = W_out^T·z(t), with W_out adapted online via a Lyapunov-derived update law Ẇ_out = −Γ·z(t)·e_x^T·P (not backpropagation).

State observer. A Luenberger-style observer augmented with the LNN's disturbance estimate: x̂̇(t) = Ax̂(t) + Bu(t) + g(x̂,t) + ζ̂_N(x̂,u) + L(y(t) − ŷ(t)).

Integral Sliding Mode Control (ISMC). The control law u = u_eq + u_nl combines an equivalent control (keeps the system on the sliding surface, using the LNN's disturbance estimate) with a discontinuous switching term (drives trajectories toward the sliding surface and rejects residual disturbance).

Stability guarantees. Two Lyapunov analyses are provided: (1) for the observer/weight-adaptation error, showing V̇_x ≤ 0 given a standard Lyapunov inequality on the observer gain L, so both the state estimation error and the LNN weight error are bounded and the state error converges to zero under persistent excitation; (2) for the sliding surface itself, showing V̇_y ≤ −η_y‖s(t)‖² for an appropriately chosen modulation gain ρ(t), guaranteeing finite-time convergence to the sliding manifold.

Experimental setup

Simulated quadrotor (mass 0.65 kg, rotor inertia 6.0×10⁻⁵ kg·m², arm length 0.26 m) tracking sinusoidal/cosinusoidal x-y reference paths (7 m amplitude) at 50 m target altitude with zero yaw. Wind disturbance modeled via the Dryden gust model. LNN observer: 32 reservoir neurons, Γ=0.03, τ=0.01, b=0.5. Benchmarked against an existing neural-network/finite-time disturbance-observer scheme.

Results

Method	Condition	MSE (x, y, z)
Existing Scheme	Without disturbance	0.015, 0.022, 0.013
Existing Scheme	With disturbance	0.028, 0.033, 0.0118
LNN (proposed)	Without disturbance	0.006, 0.009, 0.001
LNN (proposed)	With disturbance	0.009, 0.021, 0.012

The LNN-based estimator achieves consistently lower mean-square estimation error than the existing scheme across x, y, and z position states, both with and without wind disturbance — with faster convergence and accurate tracking of the combined uncertainty/disturbance term ζ_N(x,u) even under measurement noise.

Fig. 3: Estimated x-y positions under nominal (disturbance-free) conditions — proposed LNN scheme tracks the sinusoidal/cosinusoidal reference as closely as the existing scheme.

Fig. 6: Under wind disturbance, the proposed scheme (green) tracks the reference more tightly than the existing scheme (red), which shows visibly larger deviations, especially in the y-position.

Fig. 8: The LNN's estimate (orange) of the combined uncertainty/ disturbance term ζ_N(x,u) closely tracks the actual value (blue) even under significant measurement noise — this is the paper's core disturbance-estimation result.

Published: Akhtar, Z., Ijaz, S., Ahmed, Q., & Bhatti, S.G. (2026). "Liquid Neural Network-Based Integral Sliding Mode Control of Multirotor UAV under Model Uncertainty and Disturbance." 2026 American Control Conference (ACC), New Orleans, Louisiana, USA, pp. 4692–4697. (https://ieeexplore.ieee.org/abstract/document/11616285)]

Citation

@inproceedings{akhtar2026lnn,
  author    = {Akhtar, Zainab and Ijaz, Salman and Ahmed, Qadeer and Bhatti, Sidra Ghayour},
  title     = {Liquid Neural Network-Based Integral Sliding Mode Control of Multirotor UAV under Model Uncertainty and Disturbance},
  booktitle = {2026 American Control Conference (ACC)},
  year      = {2026},
  pages     = {4692--4697},
  address   = {New Orleans, Louisiana, USA}
}