Graph-Convolutional Reinforcement Learning for Finite-Time Stabilization of a Chaotic Chua Circuit via Unified Flexible Terminal Sliding-Mode Control

• Circuit-level modeling retains capacitor voltages, inductor current, and physical component parameters, making the connection between chaotic dynamics and circuit configuration explicit.Normalized models offer compact equations but partly obscure the influence of individual components.Source: §1 · p. 2
• Existing SMC approaches face tuning demands, insufficient treatment of actuator saturation, reliance on accurate dynamics, and limited use of deep or graph-based learning for parameter optimization.Source: §1 · p. 2
• The proposed graph-learning-based UFT-SMC stabilizes chaotic dynamics in finite time while accounting for lumped uncertainties, bounded disturbances, and input saturation.A GCN optimizes controller parameters according to their interdependencies and closed-loop performance.Source: §1 · p. 2
• The paper assesses the framework through convergence, control effort, integral performance indices, and comparative control studies.Source: §1 · pp. 2–3

2.1 Circuit Description and Mathematical Model

• The circuit-level model uses capacitor voltages VC1 and VC2 and inductor current iL as states, with C1, C2, L, RP and the nonlinear resistor’s physical slopes and breakpoint voltage in its formulation.The nonlinear resistor is implemented using two operational amplifiers and can have three or five linear regions, depending on resistor configuration.Source: §2.1 · p. 3

2.2 Chaotic Dynamics and Lyapunov Analysis

• At RP = 1800 Ω, the circuit exhibits irregular oscillations and an experimentally observed double-scroll attractor; the authors use that attractor to validate the chaotic operating regime.Closed-loop control results are numerical; experimental closed-loop validation is beyond the study’s scope.Source: §2.2 · p. 3, Figure 2 · p. 5
• One positive, one near-zero, and one negative Lyapunov exponent are consistent with the chaotic behavior observed in the phase-space trajectory.The Lyapunov analysis uses dimensionless time, while the control model remains in physical variables.Source: §2.2 · pp. 4–5

3.1 The control system

• The circuit-level state variables are capacitor voltages x1 and x2 and inductor current x3 in a control-oriented state-space representation.The reformulation is used to develop UFT-SMC for this system representation.Source: §3.1 · p. 6, §3 · p. 6
• The controlled model includes lumped uncertainty, external disturbances, and input saturation, with bounded uncertainty and saturation mismatch terms assumed.The nonlinear system functions are also assumed bounded as the states evolve within a bounded attractor.Source: §3.1 · pp. 6–7

3.2 The UFT-SMC control method

• The controller is designed to drive trajectories to the sliding surface despite system uncertainties and external disturbances, and the sliding variables reach it in finite time.The reaching-time bound follows from the finite-time stability argument.Source: §3.2 · pp. 8–9
• Theorem 2 establishes finite-time convergence of the Chua system to the origin under the proposed control law.After the sliding manifold is reached, the equivalent sliding dynamics drive the states to zero in finite time.Source: §3.2 · p. 9
• State-dependent exponents shape convergence in both state and sliding dynamics, while their specified ranges avoid negative-power singularities near the origin.The same exponent-shaping mechanism is used in both dynamics, giving the design a unified, structurally symmetric form.Source: §3.2 · p. 10

4 Stochastic Robustness Analysis

• The closed-loop stochastic perturbation is modeled pathwise, without requiring zero mean, finite variance, or a particular probability distribution.The analysis represents random practical effects as perturbations of the sliding dynamics and assumes measurable, locally integrable sample paths.Source: §4 · p. 10

4.1 Almost-Sure Bounded Disturbances

• The almost-sure disturbance assumption permits biased, non-Gaussian, temporally correlated perturbations, provided their sample paths satisfy the stated bound.The positive robustness margin is defined as the reaching gain minus the disturbance bound.Source: §4.1 · pp. 10–11

4.2 Almost-Sure Finite-Time Reaching

• Almost-sure finite-time reaching holds when the disturbance is bounded almost surely and the reaching gain exceeds its bound.After reaching, the sliding condition can be maintained in the Filippov sense.Source: §4.1 · p. 10, §4.2 · p. 11

4.3 Finite-Time Stabilization of the State Variables

• State stabilization remains finite-time almost surely when the robust control margin γ6 − ζ̄_i is strictly positive.The total convergence time is bounded by the sum of the sliding-manifold reaching time and the additional state convergence time.Source: §4.3 · p. 12

4.4 High-Probability Disturbance Bounds

• For a prescribed horizon, a high-probability disturbance bound yields a finite-time reaching guarantee if γ6 exceeds the bound and the reaching-time bound fits within that horizon.The pathwise proof applies on the event where the disturbance bound holds.Source: §4.4 · pp. 12–13

5.1 Learning-Based Parameter-Tuning Framework

• The learning framework optimizes controller parameters to reduce trial-and-error tuning while restricting choices to ranges consistent with the theoretical design.The objective accounts for state stabilization, convergence toward the sliding manifold, and control effort.Source: §5.1 · p. 13, §5.3 · p. 14

5.2 Graph Representation and Feature Extraction

• The GCN graph encodes the three state channels and their dynamical couplings, combining node features into a latent representation for the actor-critic.Node features include each channel’s state, sliding variable, and control input.Source: §5.2 · p. 14

5.3 Controller Parameterization and Learning Objective

• The optimization vector adjusts sliding-surface and reaching-dynamics parameters, while exponent parameters remain fixed within theoretically admissible ranges.Controller-design constraints require γj > 0 and µi > 1; the reward weights state stabilization, sliding-manifold convergence, and control effort.Source: §5.3 · p. 14

5.4 GCN-Based Actor-Critic Parameter Optimization

• A bound-preserving actor maps the latent graph representation to feasible controller parameters, and a critic estimates the long-term quality of each decision.Target networks, replay memory, temporal-difference updates, deterministic policy gradients, and exploratory perturbations support training; the best admissible parameter vector is retained after convergence.Source: §5.4 · pp. 14–15

6.1 Simulation Setup

• After activation at t = 0.1 s, the states rapidly reach a small neighborhood of the origin and remain close to zero.Before activation, the trajectories retain the uncontrolled system’s oscillatory behavior.Source: §6.1 · p. 15, Figure 6 · p. 16
• The sliding variables rapidly reach the sliding manifold within a short finite-time interval, consistent with the theoretical reaching property.Their Euclidean norm rapidly approaches zero and remains close to the origin.Source: §6.1 · pp. 16–17, Figure 7 · p. 17
• Control signals remain within the prescribed saturation limits during the transient and decrease as the states approach equilibrium.After the finite-time transient, only a small control action is required.Source: §6.1 · p. 17, Figure 8 · p. 18

6.2 Comparative and Performance Evaluation

• Compared with, UFT-SMC regulates states more rapidly and keeps them closer to zero near equilibrium, with smaller residual errors.Both controllers suppress chaotic oscillations after activation.Source: §6.2 · p. 19, Figure 9 · p. 19
• IAE decreases by 29.36%, ISE by 31.88%, ITAE by 44.56%, and ITSE by 40.38% versus.The underlying values decrease from 0.2786 to 0.1968 for IAE and from 0.1920 to 0.1308 for ISE.Source: §6.2 · p. 20

7 Conclusion

• The proposed controller drives circuit states toward the desired equilibrium in a short transient while sliding variables converge to zero and inputs stay within prescribed bounds.The design considers input saturation, uncertainties, and external disturbances.Source: §7 · p. 20
• The approach showed faster regulation, smaller residual errors, and better integral performance indices than the method used for comparison.The graph-learning scheme tunes controller parameters according to their influence on closed-loop response, reducing the need for manual adjustment.Source: §7 · p. 20
• Future work will consider experimental validation and application to other nonlinear and fractional-order dynamical systems.Source: §7 · p. 20

Declaration on the Use of Artificial Intelligence

• The authors used ChatGPT for language editing and clarity improvements and state that they remain responsible for the scientific content.Source: Declaration on the Use of Artificial… · p. 22