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.

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 Numerical Stability Test of Neutral Delay Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Z. H. Wang
Keyword(s):  
Characteristic Function ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Characteristic Root ◽  
Initial Guess ◽  
Lambert W Function ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
The Stability ◽  
Delay Differential

The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function. On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

Estimation of Time-Delays Using the Characteristic Roots of Delay Differential Equations

2011 ◽  
Author(s):  
Sun Yi ◽  
Sangseok Yu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Delay Systems ◽  
Lambert W Function ◽  
Short Paper ◽  
Characteristic Roots ◽  
General Systems ◽  
Delay Differential ◽  
Estimation Of Time

In this short paper, the preliminary result of a new method for estimation of time-delays of time-delay systems is presented. The presented method makes use of the Lambert W function, and is for scalar first-order delay differential equations (DDEs). Possible extension to general systems of DDEs and application to physical systems are also discussed.

scholarly journals On Modeling Centrifugal Compressors for Robust Control Design

1992 ◽  
Author(s):  
K. M. Murphy ◽  
D. Marchio ◽  
P. Kalata ◽  
R. Fischl
Keyword(s):  
Robust Control ◽  
Linear Model ◽  
Mass Flow ◽  
Linear Time ◽  
Control Design ◽  
Model Parameters ◽  
Inlet Valve ◽  
Linear Time Invariant ◽  
Non Linear ◽  
Robust Control Design

This paper considers the development of a Linear, Time Invariant (LTI) model for centrifugal compressor systems for robust control design. The objective is to obtain a LTI model of the compressor system so that robust controllers can be designed to control the outlet pressure and mass flow. The control variables are the motor armature voltage and the inlet valve position. The approach taken in this paper is to (1) obtain the non-linear, input-output dynamic reactions of the various components (inlet valve, ducts, compressor, nozzle and DC-motor), (2) identify the model parameters and variables (pressure, mass flow, etc.), (3) obtain a linear model of the system, (4) design a controller based on the LTI model and (5) test the designed controller on the non-linear model of the compressor system.

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