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
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/']);