Analysis and Control of Time Delayed Systems via the Lambert W Function

Design of observer-based feedback control for time-delay systems with application to automotive powertrain control

“…In this paper, it is now used to find the controller and observer gain matrices (i.e., K and L in Eqs. (10) and (11) in the next section) for the combined observer-based control in Fig. 1.…”

“…(6) is obtained numerically, for example, by using the fsolve function in Matlab; however, conditions for existence and uniqueness of such a solution to Eq. (6) are still open problems [11].…”

“…(4) is a function of A, A d , h and the preshape function, gðtÞ, and the initial condition, x 0 , while C N k is a function of A, A d , h and does not depend on gðtÞ or x 0 . The numerical and analytical methods for computing C I k and C N k were developed in [12,30]. The following condition is used to solve for the unknown matrix Q k :…”

Stability domains of the delay and PID coefficients for general time-delay systems

“…A couple of works have also employed the Nyquist plot properties to compute the delay A c c e p t e d M a n u s c r i p t 3 stability margins for the second-order and all-pole delay systems (Lee, Wang, & Xiang, 2010;Xiang, Wang, Lu, Nguyen, & Lee, 2007). The Lambert W function is used in (Yi, Nelson, & Ulsoy, 2007) to compute an analytical solution for delay differential equations. Though, numerical computations are required in some stages of the algorithm.…”

PI Controller Performance Analysis Using Lambert W Function Approach for First Order Systems with Time Delay

“…The same work was extended by R. K. Rao et al, [15] advanced the same by combining proportional, integral and velocity controllers for the same DC motor control problem with time delays. An extensive review on Lambert's approach has been given in [16] by citing various applications of its use.…”