How do I solve multivariable equation system?

Hello,

I'm trying to solve 3 equation system with 3 variables. I have implemented code presented below:

syms kp Ti Td

lambda = 5;
ksi = sqrt(2)/2;
omega = 50;

Eqs = [((0.8*Ti+0.4*kp*Ti+Td*Ti*kp)/(0.3*Ti+0.4*Td*Ti*kp) == (lambda*omega*ksi + 2*ksi*omega)),
  ((+1.5*Ti+0.4*kp+kp*Ti)/(0.3*Ti+0.4*Td*Ti*kp) == (omega^2 + 2*ksi^2*omega^2*lambda)),
  (kp/(0.3*Ti+0.4*Td*Ti*kp) == lambda*omega^3*ksi)];

[sol_kp, sol_Td, sol_Ti] =  solve(Eqs,kp,Td,Ti);

sol_kp = double(sol_kp)
sol_Ti = double(sol_Ti)
sol_Td = double(sol_Td)

ogr1 = 0.3*sol_Ti+0.4*sol_Td*sol_Ti*sol_kp;
ogr2 = 0.8*sol_Ti+0.4*sol_Ti*sol_kp+sol_Td*sol_Ti*sol_kp;
ogr3 = 1.5*sol_Ti+0.4*sol_kp+sol_Ti*sol_kp;
ogr4 = sol_kp;

s1 = ogr2/ogr1 - lambda*omega*ksi - 2*ksi*omega
s2 = ogr3/ogr1 - omega^2 + 2*ksi^2*omega^2*lambda
s3 = ogr4/ogr1 - lambda*omega^3*ksi

I obtained a solution (sol_kp, sol_Td, sol_Ti). To check whether solution is appropriate, I have computed values of s1-3 (they should be zero if the solution is correct, unfortunately they are not). I double checked equations and they seems to be right. Could you tell me if I made a mistake using the solve () function ?

Answer 1

[Torsten]

s2 = ogr3/ogr1 - omega^2 - 2*ksi^2*omega^2*Lambda

instead of

s2 = ogr3/ogr1 - omega^2 + 2*ksi^2*omega^2*Lambda

Best wishes

Torsten.

[Krzysztof Lakomy]

Thank you for your answer. That's one problem, it turned out, that solve() function swaps the order of variables. The code:

sol_kp = double(sol_Ti)
sol_Ti = double(sol_Td)
sol_Td = double(sol_kp)

causes values s1-3 be close to 0.

[Torsten]

Is this also the case if you call solve as

[sol_kp, sol_Td, sol_Ti] =  solve(Eqs,[kp,Td,Ti]);

?

Best wishes

Torsten.

[Krzysztof Lakomy]

Unfortunately, when I'm calling like this, I obtain empty variables and matlab returns warning:

Warning: 6 equations in 3 variables. 
> In C:\Program Files\MATLAB\R2012a\toolbox\symbolic\symbolic\symengine.p>symengine at 54
In mupadengine.mupadengine>mupadengine.evalin at 97
In mupadengine.mupadengine>mupadengine.feval at 150
In solve at 160
In ED_PID_synt at 20 
Warning: Explicit solution could not be found. 
> In solve at 169
In ED_PID_synt at 20