Laplace Transform of differential equations

i have a plant model with seven ODEs (cX)'=f(cX,cS,cP,Tr), (cP)'=f(cX,cS,cP), (cS)'=f(cX,cS,cP,Tr), (cO2)'=f(cX,cO2,Tr), d(Tr)'=f(Tr,Tag), d(Tag)'=f(Tr,Tag) where ' denote 1st order differential w.r.t time how do i convert to laplace domain so as to apply frequency analysis to do control design?

Answer 1

[Star Strider]

You would probably need at least the Symbolic Math Toolbox. I would need to see your actual functions (time-domain differential equations) to help you put them in a form that the Control systems Toolbox could use. The Symbolic Math Toolbox — at least the MuMath representation of it in the Symbolic Math Toolbox — is not always easy to work with. I have not delved into MuMath, so I have no idea what other capabilities it might have that the Symbolic Math Toolbox does not. My relatively straigtforward applications of it never caused problems.

Your differential equations must be linear and time-invariant with constant coefficients to be transformed into Laplace space to be used in transfer functions. If that is successful, The rest is a matter of coding. To the best of my knowledge, there are no straightforward methodS to transform Symbolic Math Laplace Transforms to the Control Systems Toolbox functions, so you must apply some ingenuity. There may be more efficient ways to do this; I‘ve not searched.

[M Gokul]

DcX=miu_x*cX*(cS/(Ks+cS))*exp(-Kp*cP)-Fe*cX/V;

DcP=miu_p*cX*(cS/(Ks1+cS))*exp(-Kp1*cP)-Fe*cP/V;

DcS=(-1/Rsx)*miu_x*cX*(cS/(Ks+cS))*exp(-Kp*cP)+(-1/Rsp)*miu_p cX(cS/(Ks1+cS))*exp(-Kp1*cP)+Fi*(cS_in/V)-Fe*cS/V;

DcO2=(k1a)*(c_O2-cO2)-rO2-Fe*cO2/V;

DTr=(Fi/V)*(Tin+273)-(Fe/V)*(Tr+273)+(rO2*deltaH)/(32*rho_r*Ch_r)- (Kt*At*(Tr-Tag))/(V*rho_r*Ch_r);

DTag=Fag*(Tin_ag-Tag)/Vj+(Kt*At*(Tr-Tag))/(Vj*rho_ag*Ch_ag); these are the ODEs that i have. I tried using dsolve command but was not working.

[Star Strider]

Please post all your code, from the syms call to the dsolve call.

[M Gokul]

this is my script file.

[Star Strider]

It works for me. Substituting this version of your dsolve call:

[cP,cX,cS,cO2,Tr,Tag] = dsolve('DcX=fcns1','DcP=fcns2','DcS=fcns3','DcO2=fcns4','DTr=fcns5','DTag=fcns6')

I get:

cP =
C6 + fcns6*t

cX =
C5 + fcns5*t

cS =
C4 + fcns4*t

cO2 =
C2 + fcns2*t

Tr =
C3 + fcns3*t

Tag =
C1 + fcns1*t

I leave it to you to make sense of all that.

[M Gokul]

is there any way to know if the output is correct? like these are nonlinear equations and i was expecting more of a complex answer!

[Star Strider]

Only a small subset of nonlinear differential equations can be solved analytically. You probably need to use one of the numeric solvers.

Code your equations as functions appropriate to the numeric solvers (see the documentation for any of them for details), then start by integrating your functions with ode45. If the solver ‘hangs’ or gives anomalous results, use ode15s instead, with the same functions to integrate your equations.

If you’re not familiar with the numeric solves, start by coding and solving simple problems to understand how to code your functions and use the solvers.

Note that the Symbolic Math Toolbox functions odeToVectorField and the matlabFunction can make the process of converting your current system of differential equations to the function form necessary to use with the numerical solvers significantly easier. Experiment with them as well.

[M Gokul]

thank you. this was a great help

[Star Strider]

My pleasure.