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Full untrimmed Flight Dynamics model of an Airplane is separated into two separate systems: Longitudinal and Lateral.
When the untrimmed model in Longitudinal direction, which is six dimensional, is trimmed (f(x,u) = 0), what is the dimension of the equilibrium space? Is it two dimensional? If yes, then why?
x = state vector for longitudinal direction u = control vector in longitudinal direction
I never heard of the term "Equilibirum Space" in flight dynamics and I can't find references in a quick google search.
However, I assume you are talking about describing the flight dynamics of an aircraft as a linear state-space model. When linearising, you can indeed separate the equations of motion in the plane symmetry (longitudinal) from the assymetric (lateral) motions.
$\dot{ \vec{x}}=\mathbf{A} \vec{x} + \mathbf{B} \vec{u} $
The dimension of the state vector $\vec{x}$ is usually 4 (velocity, flight path angle, pitch angle and pitch rate), in your case the dimension of the control vector $\vec{u}$ is 2 (thrust and elevator deflection).
You linearise the equations of motion around an equilibrium point; the state and control vectors describe the deviation from that equilibrium point. The dimension of the equilibrium points is 6 (4 state variable, 2 control variables).
One could say that the equilibrium point is in the space spanned by the 6 independent state and control variables. If that is what you mean, then dimension of the equilibrium space is 6.
The longitudinal differential system is:
$$\dot{\vec{\bf{x}}}=f(\vec{\bf{x}},\vec{\bf{u}})$$
where $\vec{\bf{x}}=[u,w,q,\theta]$, $\vec{\bf{u}}=[\delta_e,T]$
As far as linear space is concerned, control inputs $\vec{\bf{u}}$ are not states. So unless you have actuator models that construct internal actuator states, the longitudinal differential system is 4-dimensional.
As far as the equilibrium point is concerned, you set all the derivatives to zero, and impose a further restriction on $q=0$, $u=u_0$, then you have four unknowns (two state variables and two control variables) with four non-homogeneous algebraic equations. With linear aerodynamics, you have one unique solution, and it is not a vector space.
