The Lambert W function and the spectrum of some multidimensional time-delay systems

Automatica ◽  
2007 ◽  
Vol 43 (12) ◽  
pp. 2124-2128 ◽  
Author(s):  
Elias Jarlebring ◽  
Tobias Damm
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Lambert W Function ◽  
Time Delay Systems

Decay function estimation for linear time delay systems via the Lambert W function

2011 ◽  
Vol 18 (10) ◽  
pp. 1462-1473 ◽  
Author(s):  
Shiming Duan ◽  
Jun Ni ◽  
A Galip Ulsoy
Keyword(s):  
Time Delay ◽  
Decay Rate ◽  
Frequency Domain ◽  
Delay Systems ◽  
Lambert W Function ◽  
Time Delay Systems ◽  
Function Estimation ◽  
Decay Function ◽  
Optimal Decay ◽  
Optimal Decay Rate

The estimation of the decay function (i.e., [Formula: see text]; see equation (2)) for time delay systems has been a long-standing problem. Most existing methods focus on dominant decay rate (i.e., α) 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, α, 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.

Design of Observer-Based Feedback Control for Time-Delay Systems With Application to Automotive Powertrain Control

2009 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Time Delay ◽  
Feedback Control ◽  
State Feedback ◽  
Feedback Controller ◽  
State Feedback Control ◽  
Delay Systems ◽  
Lambert W Function ◽  
Actual State ◽  
Time Delay Systems ◽  
New Approach

A new approach for observer-based feedback control of time-delay systems is developed. Time-delays in systems lead to characteristic equations of infinite dimension, making the systems difficult to handle with classical control methods. In this paper, a recently developed approach, based on the Lambert W function, is used to address this difficulty and to control time-delay systems by designing an observer-based state feedback controller via eigenvalue assignment. The designed observer provides estimation of the state, which converges asymptotically to the actual state, and is then used for state feedback control. The feedback controller and the observer take simple linear forms and, thus, are easy to implement when compared to nonlinear methods. This new approach is applied, for illustration, to the control of a diesel engine to achieve improvement in fuel efficiency and reduction in emissions. The simulation results show excellent closed-loop performance.

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