function zpk and linmod

Question

Rafaella Barrêto Campos 2024-10-10

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https://ww2.mathworks.cn/matlabcentral/answers/2158895-function-zpk-and-linmod

评论： Rafaella Barrêto Campos 2024-10-16

How are matrices A,B,C,D found?

How is the calculation done to find the transfer function gains?

Example:

X_trim =

0.0156566822191464

0.0795424262493709

196.412468864823

15.6561280499959

0

-3048

U_trim =

-0.091567106993385

[A,B,C,D]=linmod(modelName,X_trim,U_trim)

A =

x1 x2

x1 -0.7302 195.5

x2 -0.1373 -0.67

B =

u1

x1 -25.91

x2 -36.5

C =

x1 x2

y1 -0.7302 195.5

y2 0 1

D =

u1

y1 -25.91

y2 0

-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)

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Answer 1

Sam Chak 2024-10-10

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https://ww2.mathworks.cn/matlabcentral/answers/2158895-function-zpk-and-linmod#answer_1530460

在 MATLAB Online 中打开

Hi @Rafaella Barrêto Campos

Once you have the state-space, you can apply the zpk() function to obtain the TF in zpk mode.

A   = [-0.7302    195.5
       -0.1373   -0.67];
B   = [-25.91
       -36.5];
C   = [-0.7302    195.5
        0         1];
D   = [-25.91
        0];
sys = ss(A, B, C, D)
sys =
 
  A = 
            x1       x2
   x1  -0.7302    195.5
   x2  -0.1373    -0.67
 
  B = 
           u1
   x1  -25.91
   x2   -36.5
 
  C = 
            x1       x2
   y1  -0.7302    195.5
   y2        0        1
 
  D = 
           u1
   y1  -25.91
   y2       0
 
Continuous-time state-space model.
G   = zpk(sys)
G =
 
  From input to output...
        -25.91 s (s+276.1)
   1:  --------------------
       (s^2 + 1.4s + 27.33)
 
         -36.5 (s+0.6327)
   2:  --------------------
       (s^2 + 1.4s + 27.33)
 
Continuous-time zero/pole/gain model.

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Rafaella Barrêto Campos 2024-10-10

Yes, how find A,B,C,D?

Sam Chak 2024-10-10

编辑：Sam Chak 2024-10-10

Hi @Rafaella Barrêto Campos

I don't know the algorithm used in the linmod.m file. However, the state-space matrices A, B, C, D are obtained by computing the Jacobian with respect to the trim condition.

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Answer 2

Aquatris 2024-10-11

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2158895-function-zpk-and-linmod#answer_1530715

Here explains how it is done. Basically you have your function that describes the relation between states and inputs. If they are nonlinear functions, you use Jacobian to linearize the relation at a particular operating point, depicted with x_trim and u_trim in your example.

Once you have a state space representation, then you can convert it to a transfer function. From the transfer function, finding the zeros poles and gain are a simple mathematical manipulation to make the transfer function look like this

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Rafaella Barrêto Campos 2024-10-11

编辑：Walter Roberson 2024-10-11

在 MATLAB Online 中打开

%% 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

Rafaella Barrêto Campos 2024-10-14

Would you like to understand how the matrices A,B,C,D are calculated?

Rafaella Barrêto Campos 2024-10-16

Loading these files generates two transfer functions, one az and the other q. I would like to know how the calculation of matrices A, B, C and D is done?

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function zpk and linmod

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