Inverse Laplace transform of a 4 by 4 matrix

Please house, I need help to output the inverse Laplace transform of the 4 by 4 matrix attached below. I will be grateful for your assistance. Thanks

Answer 1

[Sam Chak]

Hi @Sunday Aloke

Since this is a linear dynamic system described in state-space, the output can be found using the dsolve() command.

syms w(t) x(t) y(t) z(t)

%% State equations
eqns = [diff(w,t) == - 0.00022074*w +        0*x - 0.321386*y + 0.0000004924*z, 
        diff(x,t) ==   0.00012074*w - 0.187744*x + 0.321385*y +            0*z, 
        diff(y,t) ==            0*w + 0.187644*x - 0.321215*y +            0*z, 
        diff(z,t) ==            0*w +        0*x + 0.318725*y -    0.0001005*z];

%% General solution (if initial condition is unknown)
Gsol = dsolve(eqns, 'MaxDegree', 4)

Gsol = struct with fields:
    x: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481…
    w: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481…
    y: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481…
    z: C3*exp(-(t*(880740210851004153290091*((3^(1/2)*437404384016931261574875044783703658225316948059379372046244693311749352584197633693662738515710793526649883101390123898462374008216114254819…

%% Particular solution (if initial condition is known)
cond = [w(0) == 1, 
        x(0) == 0, 
        y(0) == 0, 
        z(0) == 0];
Psol = dsolve(eqns, cond, 'MaxDegree', 4)

Psol = struct with fields:
    x: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000…
    w: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000…
    y: (18429039118821823205910363600473469*(exp(-(t*(880740210851004153290091*((3^(1/2)*4374043840169312615748750447837036582253169480593793720462446933117493525841976336936627385157107935266498…
    z: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000…

[Sunday Aloke]

Thanks a million times. Let me check it

[Sam Chak]

You're welcome, @Sunday Aloke. If you find dsolve() helpful, please consider clicking 'Accept' ✔ on the answer and voting 👍 for it.

[Sunday Aloke]

I've run the code but I got an error Error in ==> "syms w(t) x(t) y(t) z(t)"

I suspected it's from my MATLAB, it hardly work when am handling any code with symbols.

The attachment is the linearized form. I want to solve the linearized form using optimal control principles.

Please, I will be glad if you can solve and drop me the solution.

[Sam Chak]

@Sunday Aloke, no worries. Enter this syntax. If the status shows "1", then it implies that the Symbolic Math Toolbox is installed in your system.

status = license('test', 'symbolic_toolbox')

status = 1

[Sunday Aloke]

I got 0 Is there anything I can do? I don't have to MATLAB CD again.

[Walter Roberson]

You are using a quite old version of MATLAB (R2007b), and your license does not include Symbolic Toolbox. It is far too late to be able to purchase the toolbox for your release.

You might be able to do something useful with a free MATLAB Online account.

There is one final possibility: R2007b was the last release that supported the older Maple-based symbolic toolbox. You just might happen to have a license for that. I would need to try to look up the licence feature name for that.

[Sam Chak]

Is it called "Maple Toolbox"? I found something here:

https://www.maplesoft.com/support/install/mtm1201_install.html

[Walter Roberson]

The historic name seems to be Symbolic Math Toolbox -- same as what it is now.

There was also the Extended Symbolic Toolbox, which gave access to the rest of Maple.

Checking, R2007b was the first release that supported the MuPAD based toolbox, but continued to support the older Maple-based toolbox if you still had it. I do not clearly recall whether it was still possible to purchase a license for the Maple-based toolbox in R2007b; my memory is telling me "No". But if you had already purchased the older toolbox for a previous release and you upgraded to R2007b (such as if you were under software support) then you would still have been able to use the Maple-based on in R2007b.

[Sunday Aloke]

@Sam Chak Please help me solve the question. I want to solve the linearized form using optimal control principles. I will copy the solution from here once solved.

Thanks

[Sunday Aloke]

@Sam Chak Please help me write the results you obtained explicitly.

Gsol = struct with fields: x: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481… w: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481… y: C1*(exp(-(t*(880740210851004153290091*((3^(1/2)*43740438401693126157487504478370365822531694805937937204624469331174935258419763369366273851571079352664988310139012389846237400821611425481… z: C3*exp(-(t*(880740210851004153290091*((3^(1/2)*437404384016931261574875044783703658225316948059379372046244693311749352584197633693662738515710793526649883101390123898462374008216114254819… %% Particular solution (if initial condition is known) cond = [w(0) == 1, x(0) == 0, y(0) == 0, z(0) == 0];

    x: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000…
    w: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000…
    y: (18429039118821823205910363600473469*(exp(-(t*(880740210851004153290091*((3^(1/2)*4374043840169312615748750447837036582253169480593793720462446933117493525841976336936627385157107935266498…
    z: (2245501681194912713180821901898316369833105182787823237505398159820541372242590057761026213554694888135433837660368384056135068619771591528250850247291290758676480000000000000000000000000…

[Sam Chak]

Hi @Sunday Aloke

The results are more intuitively readable now. If you find the solution helpful, please consider clicking 'Accept' ✔ on the answer and voting 👍 for it. Thanks a bunch!

sympref('FloatingPointOutput', true);   % <-- add this line

syms w(t) x(t) y(t) z(t)

%% State equations
eqns = [diff(w,t) == - 0.00022074*w +        0*x - 0.321386*y + 0.0000004924*z, 
        diff(x,t) ==   0.00012074*w - 0.187744*x + 0.321385*y +            0*z, 
        diff(y,t) ==            0*w + 0.187644*x - 0.321215*y +            0*z, 
        diff(z,t) ==            0*w +        0*x + 0.318725*y -    0.0001005*z];

%% Particular solution (if initial condition is known)
cond = [w(0) == 1, 
        x(0) == 0, 
        y(0) == 0, 
        z(0) == 0];
Psol = dsolve(eqns, cond, 'MaxDegree', 4)

Psol = struct with fields:
    x: 2.6046e-06*exp(-1.0001e-04*t) - 8.7541e-05*exp(-0.5090*t) + 8.4936e-05*exp(-9.6773e-05*t)*cos(0.0038*t) + 0.0202*exp(-9.6773e-05*t)*sin(0.0038*t)
    w: 5.5262e-05*exp(-0.5090*t) - 3.4039e-06*exp(-1.0001e-04*t) + 0.9999*exp(-9.6773e-05*t)*cos(0.0038*t) - 0.0253*exp(-9.6773e-05*t)*sin(0.0038*t)
    y: 8.7481e-05*exp(-0.5090*t) + 1.5220e-06*exp(-1.0001e-04*t) - 8.9003e-05*exp(-9.6773e-05*t)*cos(0.0038*t) + 0.0118*exp(-9.6773e-05*t)*sin(0.0038*t)
    z: 0.9926*exp(-1.0001e-04*t) - 5.4791e-05*exp(-0.5090*t) - 0.9925*exp(-9.6773e-05*t)*cos(0.0038*t) - 0.0065*exp(-9.6773e-05*t)*sin(0.0038*t)

w = vpa(Psol.w)

w = 

x = vpa(Psol.x)

x = 

y = vpa(Psol.y)

y = 

z = vpa(Psol.z)

z = 

[Sunday Aloke]

Accepted ✓

[Sunday Aloke]

@Sam Chak Thanks a million times. I really appreciate. Please the w=vpa(psol.w) to z=vpa(psol.z) didn't show complete.

[Sam Chak]

@Sunday Aloke, I don't understand. Can you post your code on this MATLAB Online by clicking this icon , and then hit the Run button ? This allows me to troubleshoot your code.

[Sunday Aloke]

@Sam Chak

I mean the one you solved, the answer didn't display completely.

[Sam Chak]

Hmm... This is strange. It looks complete on my machine. That's why I requested you to post exactly the same code and then hit the Run button . Let me check if I can the display completely from your side.

[Walter Roberson]

You might need to use the scroll-bars under the output expressions.

Answer 2

[Walter Roberson]

inverse laplace of a constant is dirac delta times the constant, so just multiply the matrix by dirac delta function.

Note that if your A is the A of a state space representation then you would have a completely different answer, and would need the other state space matrices as well

[Walter Roberson]

https://lpsa.swarthmore.edu/Representations/SysRepTransformations/TF2SS.html

talks about converting state space to transfer function. You would then take the inverse laplace of the transfer function.

[Sunday Aloke]

Yes, it's a state space matrix Please I am totally ignorant of the code to use. Sample of code will really help