The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function. On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.
Many engineering systems can be
modeled as neutral delay differential equations (NDDEs) that
involve a time delay in the derivative of the highest order [
The
characteristic quasipolynomial
of a DDE has an
infinite number of roots that do not have closed form, and the roots can be
found numerically only. Though the famous Newton-Raphson method works effectively in finding a root of a nonlinear equation for a
properly chosen initial guess, it does not work in finding the rightmost root of an NDDE
In
this paper, we are interested in the stability test of NDDEs,
The aim of this paper is to generalize the iterative method developed in [
A real
Let us consider a first-order
retarded delay differential equation described by
Now, if
It is not
possible to gain an explicit form of the rightmost root, as done above, for
other delay differential equations. Thus, an iterative algorithm was proposed
in [
A properly chosen initial value is
important in the applications of iterative methods. For our problem, one can
firstly chose freely a complex number
For certain fixed constants
Due to (
Such a scheme
works also for the quasipolynomial defined in (
The different
braches of Lambert
Numerical calculation of the roots of
Branch | Characteristic root | Branch | Characteristic root |
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1 | |||
2 | |||
3 | |||
4 | |||
5 | |||
10 | |||
20 | |||
100 | |||
200 | |||
0 | 500 |
The condition
In this
section, the iterative method proposed in Section
Let us
firstly consider the stability of a second-order NDDE [
To this end, one chose freely an initial guess,
say
Graphical test for the
rightmost root of (
With this
Similarly, starting from
Moreover, as shown in Figure
The solution
From
Figure
Now, let us
consider an NDDE with two delays [
Graphical test for the rightmost root
of (
Moreover, when a negative feedback
control
The plot of the real part of the rightmost
root with respect to the feedback gain
In this
paper, the iterative method based on Lambert
Though the investigation is made mainly for NDDEs with fixed parameters, the proposed scheme does work for some NDDEs with a parameter falling in a given interval. As shown in the first illustrative example from structure dynamics, for example, the iterative method can produce a plot of the real part of the rightmost root with respect to the delay, from which one can easily determine for what value of delay the system is asymptotically stable, and for what value of delay, the system is unstable. It reveals also that a proper chosen delay value can improve the stability of an NDDE. In the second illustrative example, an interval of the feedback gain is determined for stabilizing the unstable equilibrium of the NDDE by using the iterative method.
This work was supported by FANEDD of China under Grant no. 200430, and by NSF of China under Grant no. 10532050.