Hey,
The problem basically consists of an optimizing system of differential equations.
A model with varying parameters can be used to find the best-fitted model parameters for the same.
This can be achieved either through MATLAB or Simulink, both are discussed below, suitability relies upon the complexity and nature of the problem.
Following illustrates formation and solution to a differential equation.
- Using MATLAB
k1 = ...; % set constant k1
k2 = ...; % set constant k2
k3 = ...; % set constant k3
f = @(t,y)[9-k1*y(2)^2*y(1)+k2*y(1)^2 ; k3*y(2)^2+y(1)*y(2)]; % define differential equations
tspan = [0 20]; % set interval of integration
u0 = ...; % set initial condition for u
omega0 = ...; % set initial condition for Omega
y0 = [u0 ; omega0]; % set vector of initial values
[T Y] = ode45(f,tspan,y0); % call integrator
plot(t,y(:,1),'-o',t,y(:,2),'-o') % plot results
Links:
- https://www.mathworks.com/videos/solving-odes-in-matlab-8-systems-of-equations-117652.html(Differential equation formation and solution video)
- https://www.mathworks.com/help/symbolic/solve-a-single-differential-equation.html#bruif2h (Documentation)
- https://www.mathworks.com/videos/solving-odes-in-matlab-6-ode45-117537.html(Solving using ode45)
- Using Simulink
Links:
- https://www.mathworks.com/videos/simulink-a-flexible-modelling-environment-to-simulate-real-world-systems-81981.html(Modelling of differential equation video)
Optimization of differential equations
- Using MATLAB
Making use of fminsearch(), fminbnd() and optimset() functions or optimization toolbox for complex problems after solving the differential equation.
Links:
- https://www.mathworks.com/help/matlab/ref/fminsearch.html
- https://www.mathworks.com/products/optimization.html
For real-time optimization of differential equation refer to Simulink
- Using Simulink
Model can be optimized as illustrated in video
Link:
- https://www.mathworks.com/videos/optimization-of-simulink-model-parameters-81852.html(optimization of parameters video)
