Model-Reference RL Control Framework

• Model-reference reinforcement learning (MRRL) is a control framework that combines a nominal reference model with an RL agent to ensure sample efficiency and improved performance.
• The methodology integrates a baseline controller for stability with an RL-derived corrective input, achieving rapid convergence and reduced tracking errors.
• MRRL guarantees safety and convergence through Lyapunov-based methods and Bellman contraction, validated in applications like autonomous vessels and satellite formations.

A model-reference reinforcement learning (MRRL) control framework is a structured methodology that integrates conventional model-reference control principles with modern reinforcement learning (RL) architectures to achieve high-performance, sample-efficient, and provably stable control of uncertain dynamical systems. The key tenet of MRRL is to leverage a nominal reference model and a baseline controller to provide safety and stability guarantees, while an RL agent learns an additive (or multiplicative) compensation that refines tracking, adapts to uncertainties, and optimizes performance in the presence of constraints, unmodeled dynamics, or dynamic environments.

1. Mathematical Structure and Control Architecture

The MRRL framework centers on a composite closed-loop control architecture, where a nominal (reference) model defines the target behavior, a conventional baseline controller ensures local stability, and a reinforcement learning agent learns a correction for the full system's input.

Let $x(t)$ denote the true system state, $x_m(t)$ the reference model state, and $x_r(t)$ a desired trajectory. The nonlinear plant dynamics are typically of the form:

$\dot x = f(x, u) + \Delta(x, u)$

with unknown or uncertain components $\Delta(x, u)$. The reference model is designed as a simplified (often linear) system:

$\dot x_m = f_m(x_m, u_m)$

where $f_m$ is tractable and typically uncertainty-free. The desired trajectory $x_r(t)$ is generated by a motion planner or planning controller.

The baseline controller $u_b$ is synthesized to ensure $x_m(t) \rightarrow x_r(t)$. For example, in (Zhang et al., 2020), a backstepping-style tracking law is deployed for an autonomous surface vehicle:

$$u_b = D_m \nu_r + M_m(\dot \nu_r - K_\nu(\nu_m - \nu_r)) - K_\eta(\eta_m - \eta_r)$$

where $K_\eta, K_\nu$ are chosen for Hurwitz error dynamics.

The composite control command is:

$u(t) = u_b(t) + u_l(t)$

where $u_l(t)$ is the RL-learned correction. This summation structure is fundamental to MRRL (Zhang et al., 2020, Tao et al., 8 Jan 2026).

2. Reinforcement Learning Augmentation

The RL component is formulated as a Markov Decision Process (MDP), where the state typically aggregates the true state, the reference state, the baseline command, and optionally environmental or exogenous variables (e.g., obstacle positions). The action space is the corrective input $u_l(t)$.

A typical reward function penalizes both tracking error and RL control effort:

$$R_t = - (x_t - x_{m,t})^T H_1 (x_t - x_{m,t}) - u_{l,t}^T H_2 u_{l,t}$$

where $H_1, H_2 > 0$ are weighting matrices (Zhang et al., 2020, Zhang et al., 2020, Tao et al., 8 Jan 2026). Additional terms can encode safety, collision-avoidance, or domain-specific constraints.

Most MRRL implementations adopt an off-policy, actor-critic algorithm with stochastic Gaussian policies (Soft Actor-Critic is prevalent) (Zhang et al., 2020, Tao et al., 8 Jan 2026). Two multi-layer perceptrons (MLPs) serve as actor and critic, with entropy regularization providing robustness to exploration and sample efficiency.

3. Stability, Convergence, and Safety Guarantees

A defining strength of MRRL frameworks is the ability to provide formal stability and convergence guarantees, even in the presence of unmodeled dynamics and learning processes.

• Lyapunov-based stability: With an admissible baseline controller (i.e., one that renders the reference-tracking error system uniformly ultimately bounded for bounded uncertainties), it can be shown that the Lyapunov function

$V^i(e) = - Q^{\pi^i}(s, u_l^i)$

decreases at each learning iteration (Zhang et al., 2020). The improvement property,

$$\Delta V^i + (1 - \gamma) V^i + R_t^i \leq -W(e) + \mu_3(\|\Delta\|),$$

where $W$ is positive definite in the error coordinates, ensures that policy improvement progressively shrinks the deviation bound.

• Bellman contraction and convergence: For actor-critic implementations, the entropy-augmented Bellman operator is a contraction, and the sequence of soft Q-functions and policies converges to the optimal soft Q*, π* (Zhang et al., 2020).
• Practical safety: In safety-critical settings (e.g., collision-avoidance), smooth sigmoid or barrier-like terms encode risk in the reward, enabling safety to emerge as the result of optimizing expected cumulative reward rather than via hard constraints (Zhang et al., 2020, Tao et al., 8 Jan 2026).

4. Sample Efficiency and Performance Comparisons

Empirical studies consistently report that MRRL achieves significantly greater sample efficiency and superior steady-state performance relative to pure model-free RL.

• For collision-free vessel tracking, MRRL converged in approximately 200 episodes to a higher return compared with 600 episodes required by a pure RL controller, which then plateaued at a lower performance level (≈3× faster convergence) (Zhang et al., 2020).
• In high-dimensional tethered satellite formation, MRRL reduced steady-state tracking errors by over 96% (tethers) and 99% (node satellites) and decreased fuel consumption by two orders of magnitude compared to baseline-only controllers (Tao et al., 8 Jan 2026).

This efficiency arises from the baseline controller guaranteeing stability, thereby allowing the RL agent to focus its exploration on refining performance rather than avoiding catastrophic failures.

5. Integration of Constraints, Hierarchies, and Domain-Specific Extensions

MRRL frameworks can integrate complex constraints and hierarchical architectures by leveraging the structure imposed by the reference model and the modularity of the RL augmentations.

• Constraint handling: In autonomous surface vehicles, collision-avoidance is encoded as a smooth penalty based on predicted obstacle encounters, with obstacle state included in the RL observation and a barrier function in the reward (Zhang et al., 2020).
• Hierarchical training: For strongly coupled systems (e.g., tethered formation), training can be staged by first training low-level (e.g., tether) control with a fixed higher-level controller, followed by high-level (satellite) RL (Tao et al., 8 Jan 2026). This approach mitigates convergence issues arising from strong subsystem coupling.
• Reward shaping and normalization: Carefully crafted reward functions—incorporating tracking, energy use, tension, and termination criteria—combined with feature scaling (min-max normalization) further accelerate learning convergence and improve practical deployment robustness (Tao et al., 8 Jan 2026).

6. Representative Applications and Validation

MRRL frameworks have been validated across a range of complex, uncertain, and safety-critical applications:

Application Domain	Baseline Controller Type	RL Augmentation	Key Results
Autonomous surface vessels	Backstepping controller	SAC (additive correction)	3× sample efficiency; zero collision; error halving
Tethered space formation	Linear PD (2-level)	SAC (hierarchical, staged)	>96% error reduction; 100× fuel savings
General LQR (linear/nonlinear)	On-policy actor-critic	Model-reference adaptation	Uniform asymptotic stability; robust to drift

These results reflect both superior performance metrics (tracking error, constraint satisfaction) and rigorous closed-loop stability guarantees.

7. Theoretical Significance and Outlook

The MRRL paradigm establishes an overview of model-based control and reinforcement learning, enabling theoretically grounded, sample-efficient, and high-performance control under real-world uncertainty. The framework leverages Lyapunov arguments and Bellman operator properties not only for stability certification but also for learning convergence, providing a practical and rigorous pathway toward safe deployment in complex domains.

Ongoing research focuses on further generalization (e.g., to multi-agent, high-dimensional systems), integration of explicit nonlinear constraints, and principled reward shaping for safety and efficient learning, as well as application to domains such as adaptive LQR design, nonlinear chemical process control, and adaptive motion planning in unknown environments (Zhang et al., 2020, Tao et al., 8 Jan 2026, Borghesi et al., 2024).