Numerical stability analysis of a large-scale delay system modeling a lateral semiconductor laser subject to optical feedback

Verheyden, Koen; Green, Kirk; Roose, Dirk

doi:10.1103/physreve.69.036702

Cited by 8 publications

(9 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The system models a semiconductor laser subject to conventional optical feedback and lateral carrier diffusion [22]. The system in the complex scalar variable E(t), representing the electric field, and real variable N (x, t), representing the carrier density in the interval x ∈ [−0.5, 0.5], reads as…”

Section: A Semiconductor Laser Modelmentioning

confidence: 99%

See 1 more Smart Citation

A nonsmooth optimisation approach for the stabilisation of time-delay systems

et al. 2007

Self Cite

View full text Add to dashboard Cite

Abstract. This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in the design of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmost characteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precluding the use of standard optimisation methods. Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations. In dealing with systems of time-delay type, we extend the use of this algorithm to infinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms to efficiently and accurately compute the rightmost characteristic roots of such time-delay systems. Furthermore, the optimisation procedure is adapted, enabling it to perform a local stabilisation of a nonlinear time-delay system along a branch of steady state solutions. We illustrate the use of the algorithm by presenting results for some numerical examples.Mathematics Subject Classification. 65Q05, 65K10, 90C26

show abstract

Section: A Semiconductor Laser Modelmentioning

confidence: 99%

“…The functions P (x) and F (x) are specified in [22]. We split (5.7) into real and imaginary parts in order to work only with real numbers.…”

Section: A Semiconductor Laser Modelmentioning

confidence: 99%

A nonsmooth optimisation approach for the stabilisation of time-delay systems

et al. 2007

Self Cite

View full text Add to dashboard Cite

show abstract

“…Here the complex scalar variable A(t), represents the amplitude of the electrical field E(t) = A(t)e ibt , and real Z(x, t),x ∈ [−0.5, 0.5], represents the carrier density [77], The functions ζ(t), P (x) and F (x) are specified in [77]. Continuous-wave solutions, called 'external cavity modes' (ECMs) can be computed as steady-state solutions of (41)- (42), augmented with a scalar condition for the unknown b and an extra scalar constraint to remove the S 1 -symmetry.…”

Section: Dde-pde Model Of a Laser With Optical Feedbackmentioning

confidence: 99%

Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

Self Cite

View full text Add to dashboard Cite

Mathematical modeling with delay differential equations (DDEs) is widely used in various application areas of science and engineering (e.g., in semiconductor lasers with delayed feedback, high-speed machining, communication networks, and control systems) and in the life sciences (e.g., in population dynamics, epidemiology, immunology, and physiology). Delay equations have an infinite-dimensional state space because their solution is unique only when an initial function is specified on a time interval of length equal to the largest delay. Consequently, analytical calculations are more difficult than for ordinary differential equations andnumerical methods are generally the only way to achieve a complete analysis, prediction and control of systems with time delays.Delay differential equations are a special type of functional differential equation (FDE). In FDEs the time evolution of the state variable can depend on the past in an arbitrary way as long as the dependence is a bounded function of the past. However, DDEs impose a constraint on this dependence, namely that the evolution depends only on certain past values of the state at discrete times. (We do not consider here the case of distributed delay.) The delays can be constant or state dependent. The equations can also involve delayed values of the derivative of the state, which leads to equations of neutral type.In this chapter we mainly discuss the simplest case, namely a finite number of constant delays. Specifically, we consider a nonlinear system of DDEs with constant delays τ j 0, j = 1, . . . , m, of the formwhere x(t) ∈ R n , and f : R (m+1)n+p → R n is a nonlinear smooth function depending on a number of (time-independent) parameters η ∈ R p . We assume that the delays are in increasing order and denote the maximal delay by

show abstract

“…[12]. The system in the complex scalar variable A(t) and real Z(x, t), where x ∈ [−0.5, 0.5], reads as…”

Section: A Large-scale System Of Ddesmentioning

confidence: 99%

“…The functions (t), P (x) and F (x) are specified in [12]. Zero Neumann boundary conditions for Z(x, t) are imposed at x = ±0.5.…”

Section: A Large-scale System Of Ddesmentioning

confidence: 99%

Efficient computation of characteristic roots of delay differential equations using LMS methods

Verheyden

Luzyanina

Roose

2008

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

We aim at the efficient computation of the rightmost, stability-determining characteristic roots of a system of delay differential equations. The approach we use is based on the discretization of the time integration operator by a linear multistep (LMS) method. The size of the resulting algebraic eigenvalue problem is inversely proportional to the steplength. We summarize theoretical results on the location and numerical preservation of roots. Furthermore, we select nonstandard LMS methods, which are better suited for our purpose. We present a new procedure that aims at computing efficiently and accurately all roots in any right half-plane. The performance of the new procedure is demonstrated for small-and large-scale systems of delay differential equations.

show abstract

Numerical stability analysis of a large-scale delay system modeling a lateral semiconductor laser subject to optical feedback

Cited by 8 publications

References 17 publications

A nonsmooth optimisation approach for the stabilisation of time-delay systems

A nonsmooth optimisation approach for the stabilisation of time-delay systems

Continuation and Bifurcation Analysis of Delay Differential Equations

Efficient computation of characteristic roots of delay differential equations using LMS methods

Contact Info

Product

Resources

About