Summary: The estimation of the decay function (i.e., \(Ke^{\alpha t}\Phi\); see equation (2)) for time delay systems has been a long-standing problem. Most existing methods focus on dominant decay rate (i.e., \(\alpha)\) estimation, i.e., the estimation of the rightmost eigenvalue. Although some frequency domain approaches, such as bifurcation or finite dimensional approximation approaches are able to approximate the optimal decay rate computationally, the estimation of the factor, \(K\), requires knowledge of the system trajectory over time and cannot be obtained from the frequency domain alone. The existing time domain approaches, such as matrix measure/norm or Lyapunov approaches, yield conservative estimates of decay rate. Furthermore, the factor \(K\) in the Lyapunov approaches is typically not optimized.
A new Lambert W-function-based approach for estimation of the decay function for time delay systems is presented. This new approach is able to provide a closed-form solution for time delay systems in terms of an infinite series. Using this solution form, the optimal decay rate, \(\alpha\), and an estimate of the corresponding factor, \(K\), can be obtained. Less conservative estimates of the decay function can lead to more accurate description of the exponential behavior of time delay systems, and more effective control design based on those results. The method is illustrated with several examples, and results compare favorably with existing methods for decay function estimation.