Mixer

The control mixer must choose $\delta_e$ to make $\dot q=\dot q_{\mathrm cmd}$. Given $\dot q_{\mathrm cmd}$, we can determine $M_{\mathrm cmd}$ via Equation 1, and in turn $C_{M{\mathrm cmd}}$ via Equation 2.

In this simplified case, we could solve Equation 3, which is cubic in $\delta_e$, directly. However, more realistic problems will not only have more complex models, but will also be overdetermined (since there are more controls than controlled variables). Therefore, we will determine $\delta_e$ by linearizing Equation 3 about the current condition and solving that.

For the linearization, we could determine the current moment coefficient using Equation 3. But we already have an observed value of $C_M$ (which we call $C_{M_0}$), because the parameter identification requires it. In real life, $C_{M_0}$ is the output of some sort of Kalman filter; this is certain to be more accurate than the $C_M$ yielded by Equation 3, so we use it instead. We subtract this from $C_{M{\mathrm cmd}}$ to yield $\Delta C_{M{\mathrm cmd}}$.

We want $\Delta C_M$ to equal $\Delta C_{M{\mathrm cmd}}$. Expanding $\Delta C_M$ as a first-order Taylor series, we get:

$$\Delta C_M = C_M - C_{M_0} = \frac{\partial C_M}{\partial\delta_e}\Delta\delta_e + \cdots$$ (4)

We ignore all other terms and solve for $\Delta\delta_e$:

$$\Delta\delta_e = \frac{\Delta C_M}{\partial C_M/\partial \delta_e}$$ (5)

From Equation 3, we see that $\partial C_M/\partial \delta_e$ is given by:

$$\frac{\partial C_M}{\partial\delta_e} = m_2 + m_3\alpha + m_5\alpha^2$$ (6)

$$+ 2(m_4 + m_7\alpha)\delta_e + 3m_6\delta_e^2$$

The mixer uses Equations 5 and 6, with $\Delta C_{M{\mathrm cmd}}$, to determine $\delta_e$.

There is a certain advantage to this approach to a flight control system. The feedback controller does not need to concern itself with the complicated relationship between $q_{\mathrm cmd}$ and $\delta_e$; now it is only concerned with the straightforward relationship between $q_{\mathrm cmd}$ and $\dot q_{\mathrm cmd}$. It spares the feedback gains from becoming intertwined with the effectiveness of $\delta_e$ at a particular flight condition.

Operating within the feedback loop, the mixer handles the dirty work of determining $\delta_e$. The mixer provides, in effect, a gain scheduling for $\delta_e$. $1/(\partial C_M/\partial \delta_e)$ acts a lot like a gain; and varies as the effectiveness of $\delta_e$ varies.