Monthly Archives: July 2012

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.

Useful Soil Coefficients

Previously we discussed mechanical analysis and the method of sieve analysis. This allowed us to develop the particle size distribution curve. With this curve we can find some useful parameters. They are:

  • The Uniformity Coefficient
  • The Coefficient of Gradation

Both of these parameters are used in soil classification. The Uniformity Coefficient is defined as:

The Coefficient of Gradation is defined as:

where

These diameters are obtained from the particle size distribution curve by going across from the percent finer, coming to the curve, and turning 90 degrees down to the abscissa to locate the diameter as seen below.

Related Posts:

Sieve Analysis

Soil Moisture Content

Specific Gravity of Soil