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The engine as a control system

The question

What are the two numbers the controller must hold, what do they mean physically, and why can't one valve set one number?

Thrust is chamber pressure

A rocket engine burns an oxidiser and a fuel in a chamber and expands the hot gas through a converging–diverging nozzle. Thrust is momentum flux plus a pressure term at the nozzle exit (Dresia thesis eq. 2.3):

\[ F = \dot m_T\,u_e + (p_e - p_{\mathrm{amb}})\,A_e , \]

with \(\dot m_T = \dot m_{\mathrm{ox}} + \dot m_{\mathrm{fuel}}\) the total propellant flow, and \(u_e\), \(p_e\), \(A_e\) the exit velocity, pressure and area. For an ideal isentropic nozzle this becomes (eq. 2.4)

\[ F = A_t\,p_{cc}\,\sqrt{\frac{2\kappa^2}{\kappa-1}\Big(\frac{2}{\kappa+1}\Big)^{\frac{\kappa+1}{\kappa-1}}\Big[1-\Big(\frac{p_e}{p_{cc}}\Big)^{\frac{\kappa-1}{\kappa}}\Big]} + (p_e-p_{\mathrm{amb}})A_e , \]

where \(A_t\) is the nozzle throat area, \(p_{cc}\) the combustion chamber pressure and \(\kappa\) the ratio of specific heats. The throat is fixed and \(\kappa\) depends only weakly on mixture ratio, so thrust is proportional to \(p_{cc}\). That is why "thrust control becomes equivalent to \(p_{cc}\) control" (p. 11). It is also why the benchmark tracks \(p_{cc}\) rather than thrust: the test bench measures pressure directly and precisely.

Chamber pressure in turn follows the propellant flow. The classical small-signal model from Lorenzo and Musgrave, quoted in thesis eq. 2.2, is

\[ \frac{p_{cc}(s)}{\dot m_T(s)} = \frac{c^*}{A_t\,g}\;\frac{e^{-\sigma s}}{\tau s + 1}, \]

a first-order lag with dead time:

  • \(c^*\), the characteristic velocity, is a property of the propellant pair and the mixture ratio;
  • \(\sigma\) is the combustion delay;
  • \(\tau\) is the chamber filling time.

Both \(\sigma\) and \(\tau\) are short compared with everything else in a pump-fed engine. For control purposes, then: more total flow means more pressure, almost immediately. The hard part is getting the flow, which in LUMEN comes from turbopumps (The expander-bleed cycle).

Mixture ratio is temperature

The second controlled variable is the oxidiser-to-fuel mass ratio (eq. 2.1):

\[ R_{OF} = \frac{\dot m_{\mathrm{ox}}}{\dot m_{\mathrm{fuel}}} . \]

For methane, complete combustion \(\mathrm{CH_4 + 2\,O_2 \to CO_2 + 2\,H_2O}\) needs \(2 \times 32 = 64\) g of oxygen per 16 g of methane, a stoichiometric \(R_{OF}\) of 4.0. The flame is hottest near stoichiometric, so engines run fuel-rich: "operating conditions close to the stoichiometric mixture ratio needs to be avoided" to protect the chamber wall (thesis p. 11). LUMEN runs at \(R_{OF}\) 3.0–3.8, nominally 3.4 (Table 4.1).

An upward excursion is the fastest way to damage hardware. That is why classical designs make the mixture-ratio loop the fast loop and the pressure loop the slow one (Lorenzo & Musgrave 1992, p. 3; thesis p. 9).

Two valves, two outputs, no clean pairing

The coupling is built in. \(p_{cc}\) depends on the sum of the two flows and \(R_{OF}\) on their ratio, so any single valve that changes one flow moves both outputs (thesis p. 9). In LUMEN the two actions of the benchmark are the turbine valves:

  • TFV feeds the fuel turbine, which drives the fuel pump;
  • TOV feeds the oxidiser turbine, which drives the LOX pump.

DLR's static gains for a 10 % step at 40 bar (thesis Table 4.6) make the coupling concrete:

+0.1 opening of \(\Delta p_{cc}\) \(\Delta R_{OF}\)
TFV (fuel side) +3.7 bar −0.6
TOV (oxidiser side) +2.7 bar +0.9

Both valves raise pressure; they push mixture ratio in opposite directions. Holding pressure while raising \(R_{OF}\) therefore means opening TOV and closing TFV. The plane below shows this on the surrogate. Lines of constant pressure and of constant mixture ratio cross at an angle, and neither family runs parallel to a valve axis.

The usual pairing is TFV with pressure and TOV with mixture ratio. A control engineer can quantify how good it is with the relative gain array (RGA). From the gain table above, \(\lambda_{11} = g_{11}g_{22}/(g_{11}g_{22} - g_{12}g_{21}) = 3.33/(3.33 + 1.62) \approx 0.67\). That is positive and below 1: the pairing is right, but each loop sees the other one move its plant by about a third. This is the reason the PI baseline uses a static decoupler.

What it means for the agent

  • The action space is two continuous valve openings, and the outputs are a coupled, nonlinear function of both. There is no axis-aligned shortcut.
  • The two outputs are not equally dangerous. A pressure error costs thrust; a mixture-ratio error can burn the chamber. A good reward or evaluation treats them differently (From physics to reward).
  • The plant is not a lookup table. The valve plane above is the steady state. How the engine gets there takes from half a second to 20 seconds (Fast chamber, slow heat).
Check yourself (click to open)
  1. Why does the benchmark track chamber pressure and not thrust? Thrust is proportional to \(p_{cc}\) for a fixed throat, and the bench measures pressure directly and precisely.
  2. You want +2 bar at constant mixture ratio. Which valves move, and which way? Both open, TOV relatively more: TFV alone would lower \(R_{OF}\). In the valve plane, follow a mixture-ratio line upward.
  3. Why is a stoichiometric mixture ratio avoided? The flame is hottest there; the wall would overheat. LUMEN runs fuel-rich at 3.0–3.8 (stoichiometric is 4.0).