In Toolbox we realized general algorithms that
can be applied for simulation and analysis of:
- MIMO time-delay systems,
- systems with distributed and time-varying delays,
so we use only state space representation of
systems with delays and the corresponding methods
because frequency domain approach cannot be applied
to these problems.
Simulation of systems with delays is realized on the basis of the numerical Runge-Kutta-like methods [19,20,21,22].
The numerical methods are direct analogies of the
corresponding classical numerical methods of ODE theory,
i.e. if delays disappear then the methods coincide with
the classic Runge-Kutta methods for ODE.
On the basis of the numerical algorithms M-files for
time-domain analysis (time-response) of general
linear time-invariant systems with delays are elaborated.
Stability is one of the most important characteristic of (linear) systems. Toolbox contains some algorithms of verification stability of linear time-invariant systems with delays.
For linear systems with distributed and time-varying
delays there are no effective algorithmic methods
of verification the stability property.
So we propose some procedures of testing stability
property for general linear systems with delays
using numerical simulation.
LQR technique is one of the basic methods for designing stabilizing closed-loop controllers for ODEs. In Control System Toolbox [8] the effective procedure of designing linear quadratic regulator (based on solving ARE) for finite-dimensional systems is realized.
In Time-Delay System Toolbox we realized analytical methods of designing liner quadratic regulator for systems with delays [21]. The approach is the direct generalization of the corresponding classic LQR methods for ODEs, i.e. if delay disappears then the algorithms coincides with LQR algorithms of ODEs.