Performance, fragility and robustness for a class of first-order systems
with time-delay
Abstract
In recent years the use of delayed controllers has increased
considerably, since they can attenuate noise, replace derivative
actions, avoid the construction of observers, and reduce the use of
extra sensors. Also, delayed controllers have been shown to be more
insensitive to high-frequency noise. However, currently, there are few
criteria for tuning this type of controllers. This manuscript presents a
rigorous study of performance, fragility, and robustness for a
first-order system in closed-loop with a delayed controller, which leads
to consider a quasi-polynomial q( a, k, s),
where a ∈ R m denotes the system parameters, k ∈ R n are the controller
gains, and s∈C. The best performance gains k ∗ , p are obtained
for a fixed a. These gains provide the maximum exponential decay
achievable in the system response to guarantee exponential convergence
to a desired trajectory. Also, for a fixed a, criteria are given
to obtain the least fragile gains k ∗ , f that ensure the desired
trajectory tracking in the presence of controller’s gains variations.
Meanwhile, for a fixed gains k, the greatest robustness
parameters a ∗ , r are obtained. Thus, the desired trajectory tracking
of the systems is ensured in the event of parametric variations.
Finally, to illustrate and corroborate the proposed theoretical results,
a real-time implementation is presented on a mobile prototype, known as
omnidirectional mobile robot, studying a quasi-polynomial of degree 9
with three commensurable delays. These results offer convincing reasons
to implement controllers with delayed-action and take into account an
analysis of performance, fragility, and robustness to tune this type of
controllers.