**Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Materials and Methods Related to AOA Engines**

#### *Appendix A.1. Fundamental Equations of AOA Engines*

The AOA engines can be somehow compared to the A-SOFT engines since they combine two oxidizer injections in order to control the thrust and the O/F ratio [8,9]. The major difference comes from the positions of the injectors: in AOA engines, a front injection is combined with an aft-end injection (Figure 5). The swirled front injection generates the fuel regression and the thrust and the O/F ratio are then regulated by the oxidizer addition from the aft-end injection.

In AOA engines, the fuel regression rate only depends on the front injection. The swirl number provided by this injection is constant since its geometry does not change during the engine operation. The fuel regression rate can be written by Equations (A1)–(A4):

$$
\dot{r}[t] = \lg[\mathbf{G}\_o[t]],\tag{A1}
$$

*g*[*Go*[*t*]] = *a* - 1 + *S*2 *g m G<sup>n</sup> o* [*t*], (A2)

$$G\_o[t] = \frac{4}{\pi} \frac{\dot{m}\_{oFrom}[t]}{\phi^2[t]}.\tag{A3}$$

We obtain:

$$
\dot{\sigma}[t] = 4^n \pi^{-n} a \left( 1 + S\_{\mathcal{S}}^2 \right)^m \left( \frac{\dot{m}\_{\mathcal{o}From}[t]}{\phi^2[t]} \right)^n. \tag{A4}
$$

The oxidizer and the fuel mass flow rates are defined by Equations (A5) and (A6) with: *C*2 = <sup>4</sup>*<sup>n</sup>π*1−*na*(<sup>1</sup> + *S*2 *g*)*mLρ<sup>f</sup>* :

$$
\dot{m}\_0[t] = \dot{m}\_{o\,\text{Front}}[t] + \dot{m}\_{o\,A\,ft}[t],\tag{A5}
$$

$$
\dot{m}\_f[t] = \mathcal{C}\_2 \phi[t] \left( \frac{\dot{m}\_{\text{oFront}}[t]}{\phi^2[t]} \right)^n. \tag{A6}
$$

Finally, the O/F ratio can be estimated by Equation (A7):

$$\mathcal{L}[t] = \frac{\dot{m}\_{oFront}[t] + \dot{m}\_{oAft}[t]}{\mathcal{C}\_2 \phi[t] \left(\frac{\dot{m}\_{oFront}[t]}{\phi^2[t]}\right)^n}. \tag{A7}$$

The thrust can be evaluated by Equation (A8):

$$\begin{split} \, \_FF[t] &= I\_{sp} \left[ \, \_\text{S}^\text{I} \right] \, \_\text{S}^\text{O} \left\{ \dot{m}\_{o\text{Front}}[t] + \dot{m}\_{o\text{A}ft}[t] + \dot{m}\_f[t] \right\} \\ &= I\_{sp} \left[ \, \_\text{S}^\text{I} \right] \, \_\text{S}^\text{O} \left\{ \dot{m}\_{o\text{Front}}[t] + \dot{m}\_{o\text{A}ft}[t] + \text{C}\_2\phi[t] \left( \frac{\dot{m}\_{o\text{Front}}[t]}{\phi^2[t]} \right)^n \right\}. \end{split} \tag{A8}$$

Similarly to A-SOFT engines, it is possible to express the thrust and the mixture ratio as function of the three independent variables *m*˙ *oFront*, *m*˙ *oAft*, and *φ* (Equation (A9)):

$$
\begin{pmatrix} F \\ \xi^{\mathbb{F}} \end{pmatrix} = \begin{pmatrix} F \begin{bmatrix} \dot{m}\_{oFront} \, \dot{m}\_{oAft} \, \phi \end{bmatrix} \\\\ \xi^{\mathbb{F}} \begin{bmatrix} \dot{m}\_{oFront} \, \dot{m}\_{oAft} \, \phi \end{bmatrix} \end{pmatrix} . \tag{A9}
$$

*Appendix A.2. Error Propagation of AOA Engines*

The relative errors of thrust and O/F ratio in AOA engines can be written by Equations (A10) and (A11):

$$\varepsilon\_{F} = \sqrt{\left(\frac{\partial \ln F}{\partial \ln \dot{m}\_{oFrom}} \varepsilon\_{\text{fl}\_{oFrom}}\right)^{2} + \left(\frac{\partial \ln F}{\partial \ln \dot{m}\_{oAfter}} \varepsilon\_{\text{fl}\_{oAfter}}\right)^{2} + \left(\frac{\partial \ln F}{\partial \ln \Phi} \varepsilon\_{\Phi}\right)^{2}}\tag{A10}$$

$$\varepsilon\_{\xi} = \sqrt{\left(\frac{\partial \ln \zeta}{\partial \ln \dot{m}\_{oFrom}} c\_{\text{fl}\_{\text{off}}}\right)^2 + \left(\frac{\partial \ln \zeta}{\partial \ln \dot{m}\_{oAfter}} c\_{\text{fl}\_{\text{off}}}\right)^2 + \left(\frac{\partial \ln \zeta}{\partial \ln \dot{\Phi}} c\_{\Phi}\right)^2} \tag{A11}$$

Like for A-SOFT engines, we will consider that the oxidizer mass flow rate errors are equal: *em*˙ *oT* = *em*˙ *oA* = *em*˙ *oFront* = *em*˙ *oAft* = *eMFR*. The relative errors are also normalized by *eMFR*. As a consequence, the thrust, the O/F ratio and the total normalized errors for AOA are written in Equations (A12)–(A14):

$$\overline{\varepsilon\_{\rm F}} = \sqrt{\left(\frac{\partial \ln F}{\partial \ln \dot{m}\_{o\text{Front}}}\right)^2 + \left(\frac{\partial \ln F}{\partial \ln \dot{m}\_{o\text{Aft}}}\right)^2 + \left(\frac{\partial \ln F}{\partial \ln \phi}\right)^2 \overline{\varepsilon\_{\phi}}^2} \tag{A12}$$

$$\overline{\mathcal{E}\_{\xi}^{x}} = \sqrt{\left(\frac{\partial \ln \zeta^{x}}{\partial \ln \dot{m}\_{oFrom}}\right)^{2} + \left(\frac{\partial \ln \zeta^{x}}{\partial \ln \dot{m}\_{oAfter}}\right)^{2} + \left(\frac{\partial \ln \zeta^{x}}{\partial \ln \dot{\Phi}}\right)^{2} \overline{\mathcal{E}\_{\Phi}^{x}}}\tag{A13}$$

$$
\overline{\mathfrak{e}\_{total}} = \sqrt{\overline{\mathfrak{e}\_F}^2 + \overline{\mathfrak{e}\_{\xi}}^2}. \tag{A14}
$$

Similarly to A-SOFT engines, the *αT* parameter indicates the ratio of the oxidizer injections. It varies in [0, 1] (0 if: *m*˙ *oFront* = 0 and 1 if: *m*˙ *oAft* = 0) and is given by Equation (A15):

$$
\alpha\_T = \frac{\dot{m}\_{oFront}}{\dot{m}\_{oFront} + \dot{m}\_{oAft}}.\tag{A15}
$$

The sensitivity coefficients can be calculated based on Equations (A7) and (A8), and are given in the following relations (Equations (A16)–(A21)):

$$\frac{\partial \ln F}{\partial \ln \dot{m}\_{oFront}} = \frac{n + a\_T \xi}{1 + \xi} + \left( a\tau - n \right) \frac{d \ln I\_{sp}}{d \ln \xi} \,\,\,\,\tag{A16}$$

$$\frac{\partial \ln F}{\partial \ln \dot{m}\_{oAfter}} = (1 - \alpha\_T) \left\{ \frac{\xi}{1 + \xi} + \frac{d \ln I\_{sp}}{d \ln \xi} \right\},\tag{A17}$$

$$\frac{\partial \ln F}{\partial \ln \phi} = (2n - 1) \left\{ -\frac{1}{1 + \frac{\pi}{\xi}} + \frac{d \ln I\_{sp}}{d \ln \frac{\pi}{\xi}} \right\},\tag{A18}$$

$$\frac{\partial \ln \zeta}{\partial \ln \dot{m}\_{\text{oFront}}} = a\_T - n\_\prime \tag{A19}$$

$$\frac{\partial \ln \xi^{\mathbb{Z}}}{\partial \ln \dot{m}\_{oAft}} = 1 - a\_{T\prime} \tag{A20}$$

$$\frac{\partial \ln \zeta^{\mathbb{Z}}}{\partial \ln \phi} = 2n - 1. \tag{A21}$$

The specific impulse coefficient in Equations (A16)–(A18) is the same as for A-SOFT and is given in Equation (31).

#### **Appendix B. Error Propagation Results of A-SOFT Engines Operating at Non-Optimal Mixture Ratio**

*Appendix B.1. Oxidizer-Rich Combustion*

> The following case is considered:


The results are given in Figures A1 and A2.

**Figure A1.** Normalized total error (oxidizer rich combustion).

**Figure A2.** Influence of the geometric swirl number (oxidizer rich combustion) for *Sg* values in [0,2,4,6,8,10,12,15,20,30,40,50].

*Aerospace* **2019**, *6*, 65

*Appendix B.2. Fuel-Rich Combustion*

> The following case is considered:


The results are given in Figures A3 and A4.

**Figure A3.** Normalized total error (fuel rich combustion).

**Figure A4.** Influence of the geometric swirl number (fuel rich combustion) for *Sg* values in [0,2,4,6,8,10,12,15,20,30,40,50].

#### **Appendix C. Numerical Results with Downgraded Feedback Regulation Law**

**Figure A6.** Mixture ratio profile, feedback with downgraded regulation law, case 1.

**Figure A7.** Effective swirl number profile, feedback with downgraded regulation law, case 1.

**Figure A8.** Oxidizer mass flow rate profile, feedback with downgraded regulation law, case 1.

**Figure A9.** Thrust profile, feedback with downgraded regulation law, case 3.

**Figure A10.** Mixture ratio profile, feedback with downgraded regulation law, case 3.

**Figure A11.** Effective swirl number profile, feedback with downgraded regulation law, case 3.

**Figure A12.** Oxidizer mass flow rate profile, feedback with downgraded regulation law, case 3.
