how to discretize a nonlinear model using Matlab

I have this continuous nonlinear system and I want to discretize it using Matlab

If anyone can help me? This is the differential equations of the system

This is the state space model according to the authors of the paper that I’m studying

I want to know how to insert it in m files and how to make it discrete .. any idea??

Answer 1

[Alan Stevens]

This is a simultaneous, linear system.

You could do something like the following (replacing my arbitrary data with your actual data):

% Initial conditions
Te0 = 300;  Tr0 = 300;
y0 = [Te0, Tr0];

tspan = [0 10]; 
[t, y] = ode45(@rates, tspan, y0);

Te = y(:,1); Tr = y(:,2);
plot(t,Te,t,Tr)


function dydt = rates(~,y)

         % Data (obviously replace with the true values)
         Ce = 1;  Cpc = 1; Cr = 1; Cpa = 1;
         Qin = 2; Qo = 2;
         mdotr = 3;  mdotf = 3;
         en = 1;
         Tinf = 273;
         
         % Components
         Te = y(1);  Tr = y(2);
         
         % Equations
         dTedt = Qin/Ce - Cpc*mdotr*(Te - Tr)/Ce;
         dTrdt = -Qo/Cr + Cpc*mdotr*(Te - Tr)/Cr - en*Cpa*mdotf*(Te - Tinf);
         
         % Return rates
         dydt = [dTedt; dTrdt];

end

[F.M]

thank u very much.. but my main problem is how to discretize this system, your code is solving the differential equations. is there a way to disretize my system??

[Alan Stevens]

It is discretised! The solution gives discrete values of Te and Tr at discrete times.

If you want to discretise it yourself, you can set dTe/dt = (Te(i+1) - Te(i))/deltat (and a similar representation of dTr/dt), where i refers to the i'th timestep. On the right hand side you could use Te(i) and Tr(i), which would give a simple, explicit, Euler representation, or Te(i+1) and Tr(i+1), which would lead to an implicit Euler solution. Alternatively, you could use a higher order discretisation, but then you might as well use Matlab's ode45, which does just that.

It's possible I haven't fully understood what it is you want to do!

[F.M]

yes yes.. it seems right .. thank you very much..

but can I do the same using state space representation? I mean somthing like the code above? can I write a code inserting the system as matricess instead of diff equations?? what should i change?

[Alan Stevens]

You mean something like this:

 % Time data
 endtime = 10;
 N = 1000;  % number of timesteps
 dt = endtime/N;
 
 % Initial conditions
 Te0 = 300;  Tr0 = 300;
 x(:,1) = [Te0; Tr0];
 t = zeros(1,N);
 
 % Loop through timesteps
 for k = 1:N-1
     x(:,k+1) = x(:,k) + xdot(x(:,k))*dt;  % Simple Euler integration
     t(k+1) = dt + t(k); 
 end
 
 Te = x(1,:); Tr = x(2,:);
plot(t,Te,t,Tr)
 
 function dxdt = xdot(x)
     % Data
     Ce = 1;  Cpc = 1; Cr = 1; Cpa = 1;
     Qin = 2; Qo = 2;
     mdotr = 3;  mdotf = 3;
     en = 1;
     Tinf = 273;

     Te = x(1); Tr = x(2); % Extract temperatures
     
     % Define f, g and u 
     f = [Qin/Ce; -Qo/Cr];
     g = [-Cpc/Ce*(Te - Tr) 0; Cpc/Cr*(Te - Tr) -en*Cpa/Cr*(Te - Tinf)];
     u = [mdotr; mdotf];
     
     %Calculate rates of change wrt time
     dxdt = f + g*u;
 end

[F.M]

I think it may work with some modifications.. thank you very much..