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Generate system files for the indirect field oriented control system

This Jupyter Notebook generates the system files for the indirect field oriented control system given by

$$\begin{split} \begin{cases} \begin{aligned} \dot x_1 &= -c_1 x_1 + c_2 x_4 - \frac{k c_1}{u_2^0} x_2 x_4, \\ \dot x_2 &= -c_1 x_2 + c_2 u_2^0 + \frac{k c_1}{u_2^0} x_1 x_4, \\ \dot x_3 &= -c_3 x_3 - c_4 c_5 (x_2x_4 - u_2^0 x_1) + (c_4 T_m + c_3 w_{ref}), \\ \dot x_4 &= -(k_i-k_p)x_3 - k_p c_4 c_5 ( x_2 x_4 - u_2^0 x_1) + k_p (c_4 T_m + c_3 w_{ref}). \\ \end{aligned} \end{cases} \end{split}$$

Here $x_1 , x_2 , x_3$ and $x_4$ are the state variables, where $x_1$ and $x_2$ represent, respectively, direct and quadrature components of the rotor flux; $x_3$ is the rotor speed error; and $x_4$ denotes the quadrature axis component of the stator current, respectively. We also define the following constants and parameters: $u_2^0$ is a constant reference for the rotor flux magnitude; $c_1$ to $c_5$ are machine parameters; $k_p$ and $k_i$ are the proportional (P) and the integral (I) control gains, respectively; $w_{ref}$ is the speed reference; $T_m$ the load torque; $k$ the measure of rotor time constant mismatches.

These are used in the IFOC demo.

Add MatCont path and load sym package if GNU Octave is used

matcontpath = '../';
addpath(matcontpath);
addpath([matcontpath, '/Utilities']);
if isOctave
  pkg load symbolic % for GNU Octave
end

Set the system name

system_name = 'IFOC';

Create coordinates and parameter names as strings

coordsnames = {'x1', 'x2', 'x3', 'x4'};
parnames={'k', 'Tm'};

Create symbols for coordinates and parameters

The array par is the array of symbols in the same order as parnames. Due to the following two lines we may, for example, use either k or par(1). There should no changes be need of this code.

syms(parnames{:});       % create symbol for alpha and delta
par=cell2sym(parnames);  % now alpha1 is par(1) etc
syms(coordsnames{:});    % create symbol for alpha and delta
coords=cell2sym(coordsnames); % create 1 x n vector for coordinates

Define fixed parameters

c1 = 4.4868;
c2 = 0.3567;
c3 = 0;
c4 = 9.743;
c5 = 1.911;
u20 = 11.3;
kp = 4.5;
ki = 500;
wref = 0;

Define the system

dx1_dt = -c1*x1 + c2*x4 - k*c1/u20*x2*x4;
dx2_dt = -c1*x2 + c2*u20 + k*c1/u20*x1*x4;
dx3_dt = -c3*x3 - c4*c5*(x2*x4 - u20*x1) + (c4*Tm + c3*wref);
dx4_dt = -(ki-kp*c3)*x3 - kp*c4*c5*(x2*x4 - u20*x1) + kp*(c4*Tm + c3*wref);
system = [dx1_dt; dx2_dt; dx3_dt; dx4_dt];

In general there are no modifications needed after this line.

Differentiate and generate code (directional derivatives)

Exporting it to <system_name>.m. This method uses directional derivatives. Then using polarization identities derivatives can be calculated in arbitrary direction.

suc = generate_directional_derivatives(...
  system,...   % n x 1 array of derivative symbolic expressions
  coords,... % 1 x n array of symbols for states
  par,...      % 1 x np array of symbols used for parameters
  system_name,... % argument specifying the system name
  [matcontpath, 'Systems/']... % directory to save to file
);

Higher-order parameter-dependent multi-linear form.

Exporting it to <system_name>_multilinearforms.m. These multi-linear forms are currently only used in the computation of the parameter-dependent center manifold for the codimension two Bogdanov-Takens bifurcation.

order = 3;
suc = generate_multilinear_forms(system_name, system, coords, par, order, ...
        [matcontpath, 'Systems/']);

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