In this controller, time delay effect is incorporated in the mathematical model of the dynamic system throughout the control design and no approximations and assumptions are made in the controller derivation, so the system stability is easily guaranteed.
Nindjin et al. [13] incorporated time delay to system (1.2) and studied the global stability and persistence of the delayed system by using the Lyapunov functional.
In [3], Beretta and Kuang incorporated a time delay due to the gestation of the predator into system (1.1).
As to the main results in Song et al. [21], Theorem 4.1 gives a confirmative answer that there are no sustained oscillations to occur when the immune response is not incorporated with time delay.
In [9], Wang et al. incorporated a time delay into the immune response, (z'(t) = cy t-omega) - bz(t)), where the cy t-omeganse is activated at a rate proportional to the number of infected cells at a previous time, (cy(t-omega)), and it also decays exponentially at a rate proportional to its current strength bz.
In general, the lowest root-mean-square error or corrected Akaike information criterion was found with models which incorporated a time delay, defined as the fraction of maximum work or ventilation at which point a clear increase in ratings above resting level was reported.
In such equations oscillations require that explicit time delays [58] be incorporated or that the equations be elaborated to either a first-order, nonlinear, discrete-time formulation (i.e. a first-order nonlinear difference equation see [59] or [60]) or to a second-order system of differential equations.
In summary, when the model incorporated a nonlinear time delay τ ′ = τ 1 ' + τ 2 ', this delay time induces the switch from the "off" state to the "on" state.
In [6], Xu and Ma incorporated stage structure for predator and time delay due to the gestation of the predator into system (1.1).
In this paper, we have incorporated stage structure for both the predator and the prey into a predator-prey model with time delay due to the gestation of the predator.
For the estimation, we incorporated two realistic aspects of swine HEV transmission: (1) no exposure during the suckling stage and (2) time delay of seroconversion after exposure to the virus.
