It's a simple, constant coefficients, linear, first order delay equation
x'(t) = A1x(t-t1) +A2x(t-t2)+...+Amx(t-tm)+ w(t)
The Ai are constant matrices, the ti are constant delays and the w(t) is a white noise. But if you don't like the noise, it would still be helpful solving the equation without it. Also, you might want to know that I'm trying to solve the equation on a finite range [t0,t0+T].
So, can you point to any reference or tip on how to solve this?
I'm searching on books, but unfortunately they seem to be more interested on developing general conditions and theorems, than on solving this "simple" example. Or perhaps I just haven't found it yet...
EDIT
I need at least two variables, i.e. x has to be a vector with two elements for me to extract what I want from the equation. The thing is that I want to calculate the influence of one variable over the other. I can´t do that in the scalar case, but the very nice solution George Stoica pointed out in
http://www.tandfonline.com/doi/abs/10.1080/17442509208833780
Is not extensible to the vectorial case in any way I can see. If anyone has an idea on how to find the solution at least for two variables, PLEASE tell me.
Armando,
I was looking at the 1D case (where the A_i are scalars and w(t)=0) and it appears that you can solve that case using a series technique where coefficients are equated. Starting with a truncated series x(t) = sum_{n=0}^N a(n) t^n, and applying a binomial expansion, you'll end up with a linear system of equations for a(1), a(2), ..., a(N) with a(0) being an arbitrary constant. I'm thinking that this same idea can be extended to the case of arbitrary dimension.
Short of being supplied with a good reference that has already solved your problem, this is where I would go with it. But, maybe someone else has a better idea.
Gregory: thanks, it looks like a promising method to provide a suitable approximation, don´t know why restricting it to the scalar case.
I find your answer Xtremely helpful, George, I would upvote it twice if I could. I think I can do a lot with the solution for the particular case developed in the paper you pointed. Thanks a lot
Dear George,
I just don´t understand something in
http://www.tandfonline.com/doi/abs/10.1080/17442509208833780
They say the "fundamental solution" x0(t) (2.3) is defined for t>=0. However, in Eq. (2.4) and (2.5), they calculate the integral of x0(t-s-r) by s when s goes from -r to 0. If t
Dear George,
Thanks for answering me once again, I was really puzzled about this. So, just to clarify if I understand what you are telling me, I will try to rephrase it.
Clearly in (2.5) they evaluate x0 in the interval [t - r, t]. In the problematic case r > t they evaluate in a subinterval [t - r, 0]. In that subinterval you say they use the initial data, by that do you mean they set x0(t) = 1 for t < 0? In that case the integrand for those time values would be indeed just the initial function Z(s). That also makes sense because the initial function for the "fundamental solution" x0 is 1, so for t < 0 that must be the value of x0.
I'm looking forward to hear from you if I got this right.
Dear George, I realized of an issue (very fool of me not realizing before) that I included in the body of the question as an EDIT
One more time, thanks, George.
The existence and uniqueness conditions, however important, of course, are not enough for this problem I´m trying to solve. What I'm thinking now is that, starting from
dx/dt(t)=Ax(t)+Bx(t-r)+Gw(t)
And expanding x(t-r)
dx/dt(t)=Ax(t)+B( x(t) - r dx/dt(t) + o(r) ) + Gw(t)
(I+rB)dx/dt(t)=(A+B)x(t)+Bo(r)+Gw(t)
I might afford the linearization
dx/dt(t)=(I+rB)^(-1) [ (A+B)x(t)+ Gw(t)]
Assuming I+rB is inversible and r is not too large.
To sustain this, let me unveil my original problem a little further. I'm trying to make an discretized estimator for the x(t), also having some measures
zk=Hx(tk) + vk
If I propose this linearization I could then use the Kalman filter for the process, and hypothetically, correct the error caused by the linearization via the correction step of the filter. This step is meant to correct for the random errors Gw(t) and vk but it can also correct some small modelling defects, like not knowing exactly the values of the matrices and (I hope so) the errors from this linearization.
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