Trouble with inverse laplace operation

I'm trying to run a code like this:

s = tf('s')

sys = exp(-0.1*s);

sysx = pade(sys,3);

x0 = randn(10,1);

L = D-A (A,D,L are constant matrices)

H = 1/(s+L*sysx);

K = H*x0;

ilaplace(K)

Essentially, I want to get a vector x(t) in the end as per the equation xdot = -Lx(t-0.1).

But this returns the error: Undefined function 'ilaplace' for input arguments of type 'tf'.

How do I go about this? Thanks a lot.

Answer 1

[Star Strider]

You go about it with difficulty, because the Control System Toolbox and Symbolic Math Toolbox do not have any way of communicating with each other without your intervention.

Try this:

s = tf('s');
sys = exp(-0.1*s)
sysx = pade(sys,3)
sysxn = sysx.Numerator;
sysxd = sysx.Denominator;
x0 = randn(10,1);

syms A D L s t 
L = D-A;             % (A,D,L are constant matrices)

Nsysx = poly2sym(sysxn{:}, s);
Dsysx = poly2sym(sysxd{:}, s);
TFsysx = Nsysx / Dsysx;

H = 1/(s+L*TFsysx);
K = H*x0;
kh(t) = ilaplace(K, s, t);
kh(t) = vpa(kh(t), 5)

The result are a (10x1) symbolic function vector in ‘t’, ‘z’, and ‘s4’ that I will leave it to you to untangle.

Personally, I would not involve the Symbolic Math Toolbox at all, and instead evaluate the system with step, impulse, or lsim, and be happy with the result.

[Deepayan Bhadra]

Hi, thanks for your inputs. (L,D,A are constant integer matrices). When I run the code, it gives the following error while executing H = 1/(s+L*TFsysx);

Error using symengine Operands are invalid.

Error in sym/privBinaryOp (line 937) Csym = mupadmex(op,args{1}.s, args{2}.s, varargin{:});

Error in / (line 323) X = privBinaryOp(A, B, 'symobj::mrdivide');

This code worked for you, I presume? Thanks.

[Star Strider]

My pleasure.

You did not state what ‘L’, ‘D’ and ‘A’ actually were with respect to size, or modeled them as symbolic matrices. So I modeled them as symbolic scalar constants.

I simply presented an approach you can experiment with.

In all likelihood, it is not possible to do what you want. I would stay with the Control System Toolbox and not involve the Symbolic Math Toolbox at all here.

[Deepayan Bhadra]

I understand. I stuck to the Control System Toolbox and used this instead:

s = tf('s')

sys = exp(-0.1*s);

sysx = pade(sys,3);

x0 = randn(10,1);

D = diag([2 3 4 4 4 4 4 4 3 2]);

A = [0 1 1 0 0 0 0 0 0 0;1 0 1 1 0 0 0 0 0 0;1 1 0 1 1 0 0 0 0 0;0 1 1 0 1 1 0 0 0 0;0 0 1 1 0 1 1 0 0 0;0 0 0 1 1 0 1 1 0 0;0 0 0 0 1 1 0 1 1 0;0 0 0 0 0 1 1 0 1 1;0 0 0 0 0 0 1 1 0 1;0 0 0 0 0 0 0 1 1 0];

L = D-A;

H = 1/(s+L*sysx);

K = H*x0;

[y,t] = impulse(K)

The output y is null (K seems to have blown up) Somehow, this doesn't add up. I'm wondering what went wrong now.

[Star Strider]

If you do:

K = H*x0;
Num = K.Numerator
Den = K.Denominator

you will see the problem. I have no suggestions as to how to solve it.