format of the transfer function different from that contained in the articles and book

%% Find Names and Ordering of States

[sizes,x0,names]=airframe3DoF([],[],[],'sizes');
% sizes
% x0 [valores para cada names]
% names [q,theta,U,w,x,y] posicoes: q=1,theta=2, U=3, w=4, x=5, y=6.
state_names = cell(1,numel(names));
for i = 1:numel(names)
    n = max(strfind(names{i},'/')); %numero maximo de caracteres sem contar as barras.
    state_names{i}=names{i}(n+1:end);
end
%% Specify States
% Specify which states to trim and which states remain fixed.
fixed_states            = [{'U,w'} {'Theta'} {'Position'}]; % ix estados que nao mudam (nao trimam)
fixed_derivatives       = [{'U,w'} {'q'}]; % idx % wponto = az; q = thetaponto; qponto = thetapontoponto estados que quero trimar
fixed_outputs           = [];
% fixed_outputs = vazio
fixed_inputs            = [];
% fixed_inputs = vazio
n_states=[];n_deriv=[];
for i = 1:length(fixed_states)
    %length(fixed_states): tamamnho do fixed_states = 1 x 3
    n_states=[n_states find(strcmp(fixed_states{i},state_names))]; %#ok<AGROW>
    % n_states = 1 x 5; posicoes do fixed_states em relacao aos names]: [3 4 2 5 6] = [U w theta x y]
end
for i = 1:length(fixed_derivatives)
    n_deriv=[n_deriv find(strcmp(fixed_derivatives{i},state_names))]; %#ok<AGROW>
    % n_derivatives = 1 x 3; posicoes do fixed_derivatives em relacao aos names: [3 4 1] = [U w q]
end
n_deriv=n_deriv(2:end);
%  pegando os valores da segunda posicao ate a ultima, nao pega a primeira posicao, porque??? [4 1] = [w q]
%%% estava alterando o valor quando mudava o fixed_derivatives por conta da
%%% linha 71 pois ele pega do segundo valor em diante e nao pega o
%%% primeiro.
%% Trim Model
% Trim the model.
[X_trim,U_trim,Y_trim,DX,options]=trim(modelName,x0,0,[0 0 simSetup.initCond.velocityIC]', ...
    n_states,fixed_inputs,fixed_outputs, ...
    [],n_deriv)
options(10) % retorna o numero de iteracoes usadas para encontrar um ponto de corte.
%[x,u,y,dx] = trim('sys',x0,u0,y0,ix,iu,iy,dx0,idx)
% x0 = [valores iniciais para cada names]
% u0 = 0
% y0 = [0; 0; velocidadeIC]
% ix = n_states = 1 x 5; posicoes do fixed_states em relacao aos names]: [3 4 2 5 6] = [U w theta x y]
% iu = fixed_inputs = vazio
% iy = fixed_outputs = vazio
% dx0 = vazio %valores da derivada de estado no ponto inicial da busca
% idx = n_deriv pegando os valores da segunda posicao ate a ultima, nao pega a primeira posicao, porque??? [4 1] = [w q]
% idx seleciona os valores dx0 que a busca deve satisfazer exatamente.
%% Linear Model and Frequency Response
% Derive the linear model and view the frequency response.
fmt = format("longG");
% opts(3);
[A,B,C,D]=linmod(modelName,X_trim,U_trim),fmt;
% [A,B,C,D]=linmod(modelName,X_trim,U_trim),opts;
A
if exist('control','dir')
    airframe = ss(A(n_deriv,n_deriv),B(n_deriv,:),C([2 1],n_deriv),D([2 1],:));
    set(airframe,'StateName',state_names(n_deriv), ...
        'InputName',{'Elevator'}, ...
        'OutputName',[{'az'} {'q'}]);
    zpk(airframe)
    linearSystemAnalyzer('bode',airframe)
end

Function transfer:

From input "Elevator" to output...

-25.905 s (s+276.1)

az: --------------------

(s^2 + 1.4s + 27.32)

-36.503 (s+0.6328)

q: --------------------

(s^2 + 1.4s + 27.32)

Continuous-time zero/pole/gain model.

In books and articles it is stated that the equations would have this format:

a_z=(K_1 (1-s²/(ω_z^2 )))/((1+(2ζ_AF)/ω_AF s+s²/(ω_AF^2 )) )

q=(K_3 (1+T_α s))/(1+(2ζ_AF)/ω_AF s+s²/(ω_AF^2 ))

[SL: I think I formatted the equations correctly in LaTeX, though I'm not certain if the subscripts on some of the variables are supposed to be AF or if the subscript is A with F being a separate variable multiplied by that expression. If it is not correct, please clarify exactly what parts of those symbols are supposed to be subscripts and/or edit the LaTeX expressions. In case I got it wrong I left the original equations in the post.

]

The first equation "az" as shown in the model in the book and articles, in the numerador there is only contains the term s² and a constant, the term s is equal to zero. I would like to know why the term s appears? az is as an answer in the model in the numerador: ax² + bx and the correct option would be ax² + c, could someone help me please? I would like to make the equations have the same format as in articles and books, does anyone know how to solve this problem?

[Sam Chak]

@Rafaella Barrêto Campos, Do you wish to continue pursuing the explanation for transforming the transfer function into the model presented in some book and articles?