Systems of Delay Differential Equations and the Lambert W Function

The evaluation of the Lambert W function for matrices must be understood in order to move forward in our understanding of solving DDE systems. For the DDE system given as

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

The eigenvalues (in this case we will use s to denote the eigenvalues instead of the more common \lambda) of the system can be found as:

\mbox{det}\left(s{\bf I} - \dfrac{1}{\tau}W_{k}\left({\bf A}_{d}\tau\right)\right) = 0

Functions of matrices can generally be evaluated using the following relationship

F\left({\bf A}\right) = {\bf V}F\left({\bf J}\right){\bf V}^{-1}

where {\bf A} is a matrix, {\bf V} is a matrix of eigenvectors, and {\bf J} is the Jordan canonical form of the matrix {\bf A}. The Jordan canonical form of a matrix contains all the eigenvalues of the matrix and accounts for any repeated eigenvalues. As a quick example, consider a matrix {\bf A} with eigenvalues \lambda_{1} and \lambda_{2} and let’s assume that \lambda_{1} is repeated once and \lambda_{2} is repeated twice. This means that, since there are two repeated eigenvalues, there are two Jordan blocks. Then, {\bf J} can be written as

{\bf J} = \left[\begin{array}{ccccc}  \lambda_{1} & 1 & 0 & 0 & 0\\  0 & \lambda_{1} & 0 & 0 & 0\\  0 & 0 & \lambda_{2} & 1 & 0\\  0 & 0 & 0 & \lambda_{2} & 1\\  0 & 0 & 0 & 0 & \lambda_{2}  \end{array}\right] = \left[\begin{array}{cc} {\bf J}_{1} & {\bf 0}\\  {\bf 0} & {\bf J}_{2}\end{array}\right]

{\bf J}_{1} = \left[\begin{array}{cc}  \lambda_{1} & 1\\  0 & \lambda_{1}\end{array}\right]

{\bf J}_{2} = \left[\begin{array}{ccc}  \lambda_{2} & 1 & 0\\  0 & \lambda_{2} & 1\\  0 & 0 & \lambda_{2}\end{array}\right]

Each Jordan block accounts for a single repeated eigenvalue. Now, the expression F\left({\bf J}\right) is

F\left({\bf J}\right) = \left[\begin{array}{ccccc}  F\left(\lambda_{1}\right) & F'\left(\lambda_{1}\right) & 0 & 0 & 0\\  0 & F\left(\lambda_{1}\right) & 0 & 0 & 0\\  0 & 0 & F\left(\lambda_{2}\right) & F'\left(\lambda_{2}\right) & \frac{1}{2}F''\left(\lambda_{2}\right)\\  0 & 0 & 0 & F\left(\lambda_{2}\right) & F'\left(\lambda_{2}\right)\\  0 & 0 & 0 & 0 & F\left(\lambda_{2}\right)  \end{array}\right]

Thus, in general, the function of a single Jordan block (for a single repeated eigenvalue \lambda) can be written as

F\left({\bf J}\right) = \left[\begin{array}{ccccc}  F\left(\lambda\right) & F'\left(\lambda\right) & \frac{1}{2}F''\left(\lambda\right) & \cdots & \frac{1}{\left(n-1\right)!}F^{\left(n-1\right)}\left(\lambda\right)\\    0 & F\left(\lambda\right) & F'\left(\lambda\right) & \cdots & \frac{1}{\left(n-2\right)!}F^{\left(n-2\right)}\left(\lambda\right)\\    \vdots & \ddots & \ddots & \ddots & \vdots\\    {} & {} & {} & F\left(\lambda\right) & F'\left(\lambda\right)\\    0 & \cdots & \cdots & \cdots & F\left(\lambda\right)  \end{array}\right]

where n is the duplicity of the eigenvalue. If there are no repeated eigenvalues, the matrix {\bf J} becomes simple diagonal matrix with the eigenvalues on the diagonal (and zeros everywhere else). Therefore, since the Lambert W function is a function (same as sin or cos), the Lambert W of {\bf A} can be written as

W\left({\bf A}\right) = {\bf V}W\left({\bf J}\right){\bf V}^{-1}

Differentiating the defining equation x = W\left(x\right)e^{W\left(x\right)} for W and solving for W' (implicit differentiation), we obtain the following expression for the derivative of W

W'\left(x\right) = \dfrac{1}{\left(1 + W\left(x\right)\right)e^{W\left(x\right)}}
W'\left(x\right) = \dfrac{W\left(x\right)}{x\left(1 + W\left(x\right)\right)}, \mbox{ for } x\neq 0

Further derivatives of W can be taken and a general formulation of the nth derivative of W can be found in the paper “On the Lambert W Function” by Corless, et al. There is a Matlab function that will evaluate the Lambert W function for matrices (lambertwm), however you must search for it as it was written by a couple of doctoral students at Michigan.

One quick note before we move on to the current problems encountered with DDE systems. For the equation

{\bf Y}e^{\bf Y} = {\bf X},

where {\bf X} and {\bf Y} are matrices, the solution, {\bf Y} = W_{k}\left({\bf X}\right), is valid only when {\bf X} and {\bf Y} commute; i.e. when {\bf XY} = {\bf YX}. The proof is quite simple and satisfactory. First, assume that {\bf Y}e^{\bf Y} = {\bf X}. It follows that

{\bf XY} = {\bf Y}e^{\bf Y}{\bf Y} = {\bf Y}^2e^{\bf Y} = {\bf YX}

since {\bf Y} will always commute with its own matrix exponential, e^{\bf Y}. Thus, it can be seen that the two matrices satisfy the original exponential equation if and only if they commute, and thus satisfying the Lambert W function logically follows.

The next type of system that we will encounter is a DDE system with both a delay term and a non-delay term.

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

More on this to come. Stay tuned!

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 June 23, 2012, in Engineering, Linear Algebra, Mathematics. Bookmark the permalink. Leave a comment.

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