Developing and automating time delay system stability analysis of dynamic systems using the Matrix Lambert W (MLW) function method

Stability analysis of time-delayed (TD) systems is not easy to conduct since the addition of delays within a dynamic model results in irrational system equations. Traditional TD analysis methods involve adding approximations into the system model to represent these delays. Adding approximations can make the system equations rational, but will drive stable TD systems to instability as approximate accuracy is improved. A more advanced method would be an invaluable tool for simplifying the stability analysis procedure for TD systems. A new method for analyzing TD system stability without adding TD approximations to the system has been presented in the literature. This new TD stability analysis method, called the Matrix Lambert W (MLW) Function Method, involves using a matrix version of the Lambert W function to obtain analytic solutions for a set of delay differential equations. The MLW Method is discussed in five parts: (1) a fundamental understanding of the Lambert W function and the new MLW TD stability analysis method is presented; (2) a state-of-the-art review of the most current research is presented to show how the MLW Method was developed as well as the need for further development and automation of this new method; (3) a comparison of the MLW Method versus simulation is presented for three different systems: (a) a basic, time-delayed system, (b) a dynamic mechanical TD system representative of a motor grader, and (c) a metering poppet valve TD system; (4) a comparison of experimental results for the metering poppet valve versus theoretically modeled results using simulation, Pade approximations and the MLW Method is explored; (5) and finally, the system response of the simulated metering poppet valve system, called the Valve Model, obtained using an enhanced MLW Method Algorithm is presented for a range of time-delay and control-gain values. The improvements to the algorithm were achieved through the use of Valve Model validation using a new set of experimental data, phase-margin analysis and error analysis of the newly developed MLW Method Algorithm.