Eigenvalue Assignment via the Lambert W Function for Control of Time-delay Systems

Abstract

In this paper, we consider the problem of feedback controller design via eigenvalue assignment for linear time-invariant systems of linear delay differential equations (DDEs) with a single delay. Unlike ordinary differential equations (ODEs), DDEs have an infinite eigenspectrum, and it is not feasible to assign all closed-loop eigenvalues. However, we can assign a critical subset of them using a solution to linear systems of DDEs in terms of the matrix Lambert W function. The solution has an analytical form expressed in terms of the parameters of the DDE, and is similar to the state transition matrix in linear ODEs. Hence, one can extend controller design methods developed based upon the solution form of systems of ODEs to systems of DDEs, including the design of feedback controllers via eigenvalue assignment. We present such an approach here, illustrate it using some examples, and compare with other existing methods.

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

“…In [29], a numerical stabilization method was developed to obtain the rightmost eigenvalues and move them to the left half plane (LHP) using sensitivities with respect to changes in the control gain. This approach has been compared to the Lambert W function-based approach in [15]. The approach can be applied to design observers [25].…”

“…one can stabilize, improve performance, and/or meet time-domain specifications for the system (1) (e.g., see [15,22]), under the assumption that all the state variables, xðtÞ, can be measured directly. This is achieved by choosing K and K d ð2 R rÂn Þ based on desired rightmost closed-loop eigenvalues.…”

“…Spectral decomposition-based [17] Observer: an integro-differential form [18] Observer: a linear form [19] [20] Lyapunov framework [21] Linear matrix inequality (LMI) [22] Algebraic Riccati equation (ARE) [23] Coordinate transformation [24] Infinite spectrum assignment [8] Continuous pole placement [25] Finite spectrum assignment (FSA) [26] For comparisons of approaches for feedback controllers, refer to Richard [2] and Yi et al [15].…”

Back Matter

“…One can choose feedback gains to assign the rightmost pole of systems with time-delays to desired positions in the complex plane based on the Lambert W function approach (Yi et al, 2010b). Consider a simple, but unstable, example given byẋ…”

“…The Lambert W function-based approach provides a method for design of feedback controllers via pole placement (Yi et al, 2010b …”