Example of a General Nonlinear System

Physical systems where the control inputs appear nonlinearly are rather uncommon. They are, however, indispensable in the high-fidelity modeling of aircraft dynamics, where the control surfaces such as the elevator, aileron, and rudder affect the aerodynamic forces and moments of the aircraft in a nonlinear manner.

The six-degree-of-freedom model in body coordinates, where  are the inertia terms.

In[1]:=

sys = NonlinearStateSpaceModel[ {{p + r Cos[\[Phi]] Tan[\[Theta]] + 
    q Sin[\[Phi]] Tan[\[Theta]], q Cos[\[Phi]] - r Sin[\[Phi]], 
   Sec[\[Theta]] (r Cos[\[Phi]] + q Sin[\[Phi]]), 
   u Cos[\[Theta]] Cos[\[Psi]] + 
    Cos[\[Phi]] (w Cos[\[Psi]] Sin[\[Theta]] - v Sin[\[Psi]]) + 
    Sin[\[Phi]] (v Cos[\[Psi]] Sin[\[Theta]] + 
       w Sin[\[Psi]]), -w Cos[\[Psi]] Sin[\[Phi]] + (u Cos[\[Theta]] +
        v Sin[\[Theta]] Sin[\[Phi]]) Sin[\[Psi]] + 
    Cos[\[Phi]] (v Cos[\[Psi]] + 
       w Sin[\[Theta]] Sin[\[Psi]]), -u Sin[\[Theta]] + 
    Cos[\[Theta]] (w Cos[\[Phi]] + v Sin[\[Phi]]), -((
    Subscript[\[ScriptCapitalJ], \[ScriptZ]] (p q Subscript[\
\[ScriptCapitalJ], \[ScriptX]\[ScriptZ]] + 
        q r Subscript[\[ScriptCapitalJ], \[ScriptY]] - 
        q r Subscript[\[ScriptCapitalJ], \[ScriptZ]] + 
        Subscript[\[ScriptCapitalM], \[ScriptX]]) + 
     Subscript[\[ScriptCapitalJ], \[ScriptX]\[ScriptZ]] (p q \
Subscript[\[ScriptCapitalJ], \[ScriptX]] - 
        q r Subscript[\[ScriptCapitalJ], \[ScriptX]\[ScriptZ]] - 
        p q Subscript[\[ScriptCapitalJ], \[ScriptY]] + 
        Subscript[\[ScriptCapitalM], \[ScriptZ]]))/(\!\(
\*SubsuperscriptBox[\(\[ScriptCapitalJ]\), \(\[ScriptX]\[ScriptZ]\), \
\(2\)] - \(
\*SubscriptBox[\(\[ScriptCapitalJ]\), \(\[ScriptX]\)]\ 
\*SubscriptBox[\(\[ScriptCapitalJ]\), \(\[ScriptZ]\)]\)\))), (-p r \
Subscript[\[ScriptCapitalJ], \[ScriptX]] + (-p^2 + 
       r^2) Subscript[\[ScriptCapitalJ], \[ScriptX]\[ScriptZ]] + 
    p r Subscript[\[ScriptCapitalJ], \[ScriptZ]] + 
    Subscript[\[ScriptCapitalM], \[ScriptY]])/
   Subscript[\[ScriptCapitalJ], \[ScriptY]], (p q 
\!\(\*SubsuperscriptBox[\(\[ScriptCapitalJ]\), \(\[ScriptX]\), \(2\)]\
\) + Subscript[\[ScriptCapitalJ], \[ScriptX]\[ScriptZ]] (p q \
Subscript[\[ScriptCapitalJ], \[ScriptX]\[ScriptZ]] + 
       q r Subscript[\[ScriptCapitalJ], \[ScriptY]] - 
       q r Subscript[\[ScriptCapitalJ], \[ScriptZ]] + 
       Subscript[\[ScriptCapitalM], \[ScriptX]]) + 
    Subscript[\[ScriptCapitalJ], \[ScriptX]] (-q r Subscript[\
\[ScriptCapitalJ], \[ScriptX]\[ScriptZ]] - 
       p q Subscript[\[ScriptCapitalJ], \[ScriptY]] + 
       Subscript[\[ScriptCapitalM], \[ScriptZ]]))/(-
\!\(\*SubsuperscriptBox[\(\[ScriptCapitalJ]\), \(\[ScriptX]\[ScriptZ]\
\), \(2\)]\) + 
    Subscript[\[ScriptCapitalJ], \[ScriptX]]
      Subscript[\[ScriptCapitalJ], \[ScriptZ]]), 
   r v - q w + (\[ScriptCapitalT] + \[ScriptCapitalX] - 
     g m Sin[\[Theta]])/m, -r u + p w + \[ScriptCapitalY]/m + 
    g Cos[\[Theta]] Sin[\[Phi]], 
   q u - p v + \[ScriptCapitalZ]/m + 
    g Cos[\[Theta]] Cos[\[Phi]]}, {\[Phi], \[Theta], \[Psi], x, y, z, 
   p, q, r, u, v, w}}, {\[Phi], \[Theta], \[Psi], x, y, z, p, q, r, u,
    v, w}, {Subscript[\[ScriptCapitalM], \[ScriptX]], 
    Subscript[\[ScriptCapitalM], \[ScriptY]], 
    Subscript[\[ScriptCapitalM], \[ScriptZ]], \[ScriptCapitalX], \
\[ScriptCapitalY], \[ScriptCapitalZ]}, {Automatic, Automatic, 
   Automatic, Automatic, Automatic, Automatic, Automatic, Automatic, 
   Automatic, Automatic, Automatic, Automatic}, Automatic , 
   SamplingPeriod -> None];

The longitudinal dynamics are the dynamics for states , , , and :