Feedback Control Via Eigenvalue Assignment for Time Delayed Systems Using the Lambert W Function

2007 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Controller Design ◽  
Analytical Form ◽  
Lambert W Function ◽  
Feedback Controllers ◽  
Delayed Systems ◽  
Linear Delay ◽  
The Matrix ◽  
Delay Differential ◽  
Eigenvalue Assignment

In this paper, we consider the problem of feedback controller design via eigenvalue assignment for systems of linear delay differential equations (DDEs). 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 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 ODEs to systems of DDEs, including the design of feedback controllers via eigenvalue assignment. We present such an approach here, illustrate using some examples, and compare with other existing methods.

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

2010 ◽  
Vol 16 (7-8) ◽  
pp. 961-982 ◽  
Author(s):  
S. Yi ◽  
P.W. Nelson ◽  
A.G. Ulsoy
Keyword(s):  
Differential Equations ◽  
Controller Design ◽  
Linear Time ◽  
Analytical Form ◽  
Delay Systems ◽  
Lambert W Function ◽  
Linear Time Invariant ◽  
Linear Delay ◽  
Delay Differential ◽  
Eigenvalue Assignment

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.

scholarly journals On the Convergence of the Matrix Lambert W Approach to Solution of Systems of Delay Differential Equations

2018 ◽  
Author(s):  
A. Galip Ulsoy ◽  
Rita Gitik
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Iteration Method ◽  
Delay Differential Equations ◽  
Function Method ◽  
Lambert W Function ◽  
The Matrix ◽  
Iterative Solutions ◽  
Convergence Problems ◽  
Delay Differential

Convergence aspects of the matrix Lambert W function method for solving systems of delay differential equations (DDEs) are considered. Recent research results show that convergence problems can occur with certain DDEs when using the well-established Q-iteration approach. A complementary, and recently proposed, W-iteration approach is shown to converge even on systems where the Q-iteration fails. Furthermore, the role played by the branch numbers k = -∞ .. -2, -1, 0, 1, 2 , .. ∞ of the matrix Lambert W function, Wk, in terms of initializing the iterative solutions, is also discussed and elucidated. Several second order examples, known to have convergence problems with Q-iteration, are readily solved by W-iteration. Examples of third and fourth order DDEs show that the W-iteration method is also effective on higher-order systems.

scholarly journals On the Convergence of the Matrix Lambert W Approach to Solution of Systems of Delay Differential Equations

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
A. Galip Ulsoy ◽  
Rita Gitik
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Function Method ◽  
Higher Order ◽  
Lambert W Function ◽  
The Matrix ◽  
Iterative Solutions ◽  
Convergence Problems ◽  
Delay Differential

Abstract Convergence of the matrix Lambert W function method for solving systems of delay differential equations (DDEs) is considered. Recent research shows that convergence problems occur with certain DDEs when using the well-established Q-iteration approach. A complementary, and recently proposed, W-iteration approach is shown to converge even on systems where Q-iteration fails. Furthermore, the role played by the branch numbers k = −∞ … −1, 0, 1, … ∞ of the matrix Lambert W function, Wk, in terms of initializing the iterative solutions, is also discussed and elucidated. Several second-order examples, known to have convergence problems with Q-iteration, are readily solved by W-iteration. Examples of third- and fourth-order DDEs show that W-iteration is also effective on higher-order systems.

Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis

2006 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Chatter Stability ◽  
Approximate Methods ◽  
Linear Delay ◽  
The Matrix ◽  
The Stability ◽  
Delay Differential

We investigate the stability of the regenerative machine tool chatter problem, in a turning process modeled using delay differential equations (DDEs). An approach using the matrix Lambert function for the analytical solution to systems to delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert function, known to be useful for solving scalar first order DDEs, has recently been extended to a matrix Lambert function approach to solve systems of DDEs. The essential advantage of the matrix Lambert approach is not only the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy, and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations via Eigenvalue Assignment

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Robust Control ◽  
Differential Equations ◽  
Time Domain ◽  
Delay Differential Equations ◽  
Linear Time ◽  
Lambert W Function ◽  
Linear Time Invariant ◽  
Robust Control Design ◽  
Delay Differential ◽  
Eigenvalue Assignment

An approach to eigenvalue assignment for systems of linear time-invariant (LTI) delay differential equations (DDEs), based upon the solution in terms of the matrix Lambert W function, is applied to the problem of robust control design for perturbed LTI systems of DDEs, and to the problem of time-domain response specifications. Robust stability of the closed-loop system can be achieved through eigenvalue assignment combined with the real stability radius concept. For a LTI system of DDEs with a single delay, which has an infinite number of eigenvalues, the recently developed Lambert W function-based approach is used to assign a dominant subset of them, which has not been previously feasible. Also, an approach to time-domain specifications for the transient response of systems of DDEs is developed in a way similar to systems of ordinary differential equations using the Lambert W function-based approach.

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