Solid-Fuel Ramjet: Principles & Control

• Solid-Fuel Ramjets (SFRJs) are airbreathing propulsion systems that use a combustion chamber lined with solid fuel undergoing pyrolysis to generate thrust at high speeds.
• High-fidelity CFD and reduced-order models solve compressible Navier-Stokes equations with additional source terms to capture complex interactions among fluid dynamics, combustion, and fuel regression.
• Adaptive control techniques, including RCAC and DMAC, along with neural network-based estimators, enable robust thrust regulation despite strong nonlinearities and dynamic port geometry changes.

A solid-fuel ramjet (SFRJ) is an airbreathing propulsion system that generates thrust by combusting solid fuel with atmospheric air, utilizing the ram effect at high speeds to compress the incoming airflow. Unlike hybrid or liquid-fuel ramjets, the SFRJ features a combustion chamber lined with solid fuel that regresses via pyrolysis and oxidizes with onboard or ingested air. SFRJs offer a compact, energy-dense, throttling-capable propulsion option for long-range, high-speed applications but are characterized by strong nonlinearities and multi-physics coupling among fluid dynamics, combustion, and fuel regression processes, making thrust regulation challenging (Khokhar et al., 6 Nov 2025).

1. Physical Principles and Modeling of SFRJ

The SFRJ combustor is fundamentally governed by the multispecies, compressible Navier–Stokes equations with additional source terms for pyrolysis and combustion. In high-fidelity computational studies, such as those employing SU2‐NEMO, the axisymmetric, reacting Navier–Stokes equations are solved in thermochemical equilibrium:

• Continuity:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$

• Momentum:

$$\frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u}\otimes \mathbf{u} + p \mathbf{I}) = \nabla \cdot \boldsymbol{\tau}$$

where $\boldsymbol{\tau}$ is the viscous stress tensor.

• Total Energy:

$$\frac{\partial E}{\partial t} + \nabla \cdot \bigl((E + p)\,\mathbf{u}\bigr) = \nabla \cdot (k \nabla T) + \boldsymbol{\tau}:\nabla \mathbf{u} + \dot{Q}$$

with $E = \rho(e + \frac{1}{2}\|\mathbf{u}\|^2)$.

• Species Continuity for each $s$:

$$\frac{\partial (\rho Y_s)}{\partial t} + \nabla \cdot (\rho \mathbf{u} Y_s) = -\nabla\cdot \mathbf{j}_s + \dot{\omega}_s$$

Fuel regression is typically modeled by empirical laws relating the regression rate $\dot r$ to local air mass flux $G$, chamber pressure, and temperature:

$\dot r = \alpha G_a^a P_4^b T_{t2}^c$

with $G_a$ denoting the local air mass flux at the fuel port, and coefficients determined experimentally (Oveissi et al., 4 Jan 2026, DeBoskey et al., 9 Jun 2025). The evolving port geometry due to burning solid fuel creates a dynamically changing combustion chamber and coupled flowpath.

In reduced-order modeling, quasi-one-dimensional conservation laws are used:

• Mass:

$$\frac{\partial}{\partial t}(\rho A) + \frac{\partial}{\partial x}(\rho u A) = \dot m_{\text{fuel}}'(x,t)$$

• Momentum:

$$\frac{\partial}{\partial t}(\rho u A) + \frac{\partial}{\partial x}[(\rho u^2 + p)A] = p\,\frac{dA}{dx}$$

• Species and Energy as above.

The net thrust is then predicted via a control-volume analysis:

$$T = \dot m_{\text{air}} (1+f) u_e - \dot m_{\text{air}} u_\infty$$

where $f$ is the fuel–air ratio, $u_e$ is exhaust velocity, and $u_\infty$ is freestream velocity (DeBoskey et al., 9 Jun 2025, Oveissi et al., 4 Jan 2026).

2. Operational Envelope, Combustion Limits, and Instability

The SFRJ operational regime is constrained by limits on fuel regression, combustion stability, and flow choking:

• Blowout Limit: Insufficient fuel mass injection leads to combustion extinction.
• Thermal Choking/unstart: Excessive heat and fuel flow cause the combustor Mach number to approach unity, driving a backpressure that disrupts inlet shock systems and collapses thrust ("inlet unstart"). This occurs at a critical nondimensional heat flux $K_h^*$, identified both analytically and computationally (Khokhar et al., 6 Nov 2025).
• Geometry and Pressure Coupling: Regression widens the port, altering flowfields and residence times, further complicating control.

High-fidelity CFD can resolve the steady-state thrust map for SFRJs by varying Mach number, inlet total–static pressure ratio, fuel geometry, and heat flux. Diagnosis of regime boundaries is achieved using flowfield Mach and pressure contours: pre-unstart, the combustor core operates subsonically; beyond $q_h^*$, a sudden drop in thrust and flow reversal occurs.

3. Sensing and Model Reduction for Thrust Estimation

Direct measurement of thrust in flight is generally impractical. Research has established the use of neural networks (NNs) trained on synthetic data from physics-based simulations to estimate thrust using a minimal set of in-situ sensors (DeBoskey et al., 9 Jun 2025, Oveissi et al., 4 Jan 2026). Typical sensors include:

• Nozzle pressure and velocity (for DMAC frameworks).
• Cowl/inlet geometry, combustor pressure, exhaust CO mole fraction, altitude (for NN-based estimation).

A representative NN estimator is:

• Input: $[r_0, P_{t4}, X_{CO}, H]$ (cowl radius, total pressure, CO mole fraction, altitude),
• Architecture: one hidden layer, 20 neurons, sigmoid activation,
• Output: $\hat{T}$ (predicted thrust).

Training is conducted on $10^5 +$ synthetic simulation runs, using mean squared error as loss (DeBoskey et al., 9 Jun 2025).

4. Adaptive Thrust Control Techniques

SFRJ thrust regulation leverages adaptive, data-driven controllers to overcome model uncertainty and strong nonlinearities. Recent methods include:

• Retrospective Cost Adaptive Control (RCAC): An adaptive proportional–integral (PI) or PID law is applied, with online gain updates performed through recursive least squares minimization of a retrospective cost function that penalizes lagged tracking error and deviation from nominal gains (Khokhar et al., 6 Nov 2025, DeBoskey et al., 9 Jun 2025). The control command is typically either cowl translation (modulating air capture) or heat-flux injection at the grain surface.
• Dynamic Mode Adaptive Control (DMAC): DMAC constructs an online, local, linear approximation of the SFRJ system via dynamic mode decomposition (DMD) combined with recursive least squares (RLS). States $\xi_k$ (pressure, velocity, or augmented vectors) and control $u_k$ are used to iteratively fit affine state-space models

$\xi_{k+1} = A_k \xi_k + B_k u_k$

The control law augments full-state feedback with integral error and exploration noise:

$u_k = K_{\xi,k} \xi_k + K_{q,k} q_k + v_k$

Gains $(K_{\xi,k}, K_{q,k})$ are computed by discrete-time Linear–Quadratic–Integral (LQI) design around the identified $(A_k, B_k)$, ensuring Lyapunov stability under standard conditions (Oveissi et al., 4 Jan 2026, Oveissi et al., 4 Jan 2026).

• Integral Action and Persistent Excitation: All effective schemes enforce zero steady-state error via integral action and embed probing (usually Gaussian) noise to maintain model identifiability (persistent excitation), particularly critical in recursive identification settings (Oveissi et al., 4 Jan 2026).

5. Closed-Loop Performance and Robustness

Extensive simulation studies across high-fidelity and reduced-order models quantify tracking performance, transient response, and robustness:

• Tracking Error: DMAC and RCAC controllers drive steady-state tracking error $|y_k-r_k|$ below 1% of reference thrust, with typical settling times under 0.1 s (Oveissi et al., 4 Jan 2026, DeBoskey et al., 9 Jun 2025).
• Disturbance Rejection: Step, doublet, ramp, and random thrust commands are accurately followed without retuning control parameters, and controllers exhibit insensitivity to rapid changes in altitude, Mach number, or inlet conditions (Khokhar et al., 6 Nov 2025).
• Hyperparameter Sensitivity: Performance degrades gracefully and remains within operational tolerances even for order-of-magnitude variations in critical hyperparameters (RLS regularization matrices, LQR weights, NN hidden units). In DMAC, $\max_k |z_k|$ remains $\lesssim$ 1 N for all tested configurations (Oveissi et al., 4 Jan 2026).
• Failure Modes: At very low-thrust or low-altitude extremes, the local linearization may leave the excitation region, leading to control degradation in model-free schemes (Oveissi et al., 4 Jan 2026).

The following table summarizes closed-loop metrics from key studies:

Reference	Control Method	Steady-state Error	Settling Time	Robustness Considerations
(Oveissi et al., 4 Jan 2026)	DMAC	< 1 N tracking error	< 50 steps	Robust across $R_\Theta$, $R_1$, $R_2$ variations
(DeBoskey et al., 9 Jun 2025)	RCAC PID	< 1%	< 0.1 s	Stable for $p$ varied $10^2\times$
(Khokhar et al., 6 Nov 2025)	RCAC PI	$\pm5$ N (600 N cmd)	50–75 ms	Robust to 15 random inlet scenarios
(Oveissi et al., 4 Jan 2026)	DMAC	< 1%	< 0.1 s	97.5% success (200 MC runs)

6. Implications for Design and Future Research

The convergence of high-fidelity CFD, physics-guided reduced-order modeling, and data-driven adaptive control has enabled robust, model-free thrust regulation in SFRJs. Key implications:

• Minimal Instrumentation: Only a small set of in-situ sensors (e.g., nozzle pressure, velocity, combustion product mole fractions) are required; explicit analytical SFRJ models are not necessary for effective closed-loop control (Oveissi et al., 4 Jan 2026).
• Operational Flexibility: Adaptive, learning-based controllers automatically accommodate the strongly time-varying and nonlinear SFRJ operating regime, including port regression and fuel depletion.
• Hyperparameter Robustness: Graceful performance degradation under large hyperparameter variations enables practical hardware deployment without extensive manual retuning.
• Limitation: All approaches depend on sufficient measurement persistency; insufficient excitation or sensor dropout may undermine identifiability and stability.

This suggests that SFRJ systems combined with model-free adaptive control are technically viable for robust thrust modulation in future airbreathing propulsion applications, particularly where analytical reduced-order models are prohibitively complex or unavailable (Khokhar et al., 6 Nov 2025, Oveissi et al., 4 Jan 2026, DeBoskey et al., 9 Jun 2025, Oveissi et al., 4 Jan 2026).