Systems of Delay Differential Equations and the Lambert W Function Continued…

Now we reach the end of this particular journey down the road of DDE systems. As it turns out the problem of determining the stability of a DDE system with both a time dependent term and a delay term is very difficult. Once again consider the DDE system

\dot{\bf x}\left(t\right) = {\bf Ax}\left(t\right) + {\bf A}_d {\bf x}\left(t-\tau\right)

The method that we will use to solve this system is to use the Laplace transform with zero initial conditions.

s{\bf X}\left(s\right) = {\bf AX}\left(s\right) + {\bf A}_d {\bf X}\left(s\right)e^{-s\tau}

Once again, we divide out {\bf X}\left(s\right), rearrange, and are left with

\left(s{\bf I} - {\bf A}\right)e^{s\tau} = {\bf A}_d

In order to get this into the form necessary to use the Lambert W function, we multiply both sides by \tau e^{-{\bf A}\tau}.

\left(s{\bf I} - {\bf A}\right)e^{s\tau}\tau e^{-{\bf A}\tau} = {\bf A}_d \tau e^{-{\bf A}\tau}

\left(s{\bf I} - {\bf A}\right)\tau e^{\left(s{\bf I} - {\bf A}\right)\tau} = {\bf A}_d \tau e^{-{\bf A}\tau}

With our current knowledge of the Lambert W function, we have this problem pegged, right? We can use the Lambert W function to solve this equation easily.

s{\bf I} = \dfrac{1}{\tau}W_{k}\left({\bf A}_d \tau e^{-{\bf A}\tau}\right) + {\bf A}

This means the eigenvalues of the system can be found as

\mbox{det}\left(s{\bf I} - \left[\dfrac{1}{\tau}W_{k}\left({\bf A}_d \tau e^{-{\bf A}\tau}\right) + {\bf A}\right]\right) = 0

However, there is a problem with this conclusion. Remember our discussion on how matrices must commute (i.e. {\bf XY} = {\bf YX}) in order for them to satisfy the original exponential equation leading to the Lambert W function? Look again at this equation.

\left(s{\bf I} - {\bf A}\right)\tau e^{\left(s{\bf I} - {\bf A}\right)\tau} = {\bf A}_d \tau e^{-{\bf A}\tau}

Well it turns out that, in general, the matrices \left(s{\bf I} - {\bf A}\right)\tau and {\bf A}_d \tau e^{-{\bf A}\tau} do not commute. In other words, in general,

\left[\left(s{\bf I} - {\bf A}\right)\tau\right]\left[{\bf A}_d \tau e^{-{\bf A}\tau}\right] \neq \left[{\bf A}_d \tau e^{-{\bf A}\tau}\right]\left[\left(s{\bf I} - {\bf A}\right)\tau\right] (it’s true, run some numbers!!)

This is due to the fact that,for a general case, {\bf AA}_d \neq {\bf A}_d{\bf A}

As is turns out, there is a proposed method to deal with this problem which involves introducing an unknown matrix into the equation that forces commutivity. Thus, we will explore this in the next, and final, post.

About Jordan Jameson

I am a PhD mechanical engineering student at the University of Maryland College Park. I am performing research at the Center for Advanced Life Cycle Engineering (CALCE) in the area of Prognostics and Health Management (PHM) and physics of failure (POF). I enjoy science and discovery, especially the applications to engineering. Follow me on Twitter @NJordanJameson.

Posted on July 18, 2012, in Engineering, Linear Algebra, Mathematics. Bookmark the permalink. Leave a comment.

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