Exemplo de um sistema não linear geral

Sistemas físicos onde os inputs de controle parecem não lineares são relativamente incomuns. Eles são, contudo, indispensáveis na modelagem de alta fidelidade de dinâmica de aeronaves, onde as superfícies de controle, como o elevador, aileron e leme, afetam as forças e momentos aerodinâmicos da aeronave de forma não linear.

O modelo de seis graus de liberdade em coordenadas do corpo, onde  são os termos de inércia.

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];

As dinâmicas longitudinais são as dinâmicas para os estados , , , e :