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Extra resources for Applications of Time Delay Systems
Nihtil¨ a M. (1991) Pole placement design methodology for input delay systems. In European Control Conference, Grenoble, France 16. J. (2001) Delay robustness of closed loop ﬁnite assignment for input delay systems. In Proc. of the 3rd IFAC Workshop on Time Delay Systems, Santa Fe, New Mexico 17. , and Fujita M. (2003) Predictive H ∞ control for linear systems over communication channels with time-varying delays. In Proc. of the 4th IFAC Workshop on Time Delay Systems, Rocquencourt, France 18. , Kojima A.
However, assuming that we have the time stamps of the previous control signals applied to the plant, we can obtain an estimate of the next controller-to-plant time-delay. So, considering that this prediction of τcp is accurate with some degree of conﬁdence, we can estimate the plant’s state at the time of arrival of the control signal. Proceeding in a similar fashion to the previous subsection, the state of the system in equation (3), in the interval kts ≤ t ≤ (k + 1)ts , can be obtained in the following steps: x[kts + τpc ] = Φ3 x[kts ] + Γ3 x[(k − 1)ts + τpc + τcp ] x[kts + τpc + τcp ] = Φ4 x[kts + τ pc] + Γ4 x[(k − 1)ts + τpc + τcp ] x[(k + 1)ts ] = Φ5 x[kts + τpc + τcp ] + Γ5 x[kts + τpc + τcp ] x[(k + 1)ts + τpc ] = Φ6 x[kts + τpc + τcp ] + Γ6 x[kts + τpc + τcp ] x[(k + 1)ts + τpc + τcp ] = Φ7 x[kts + τpc + τcp ] + Γ7 x[kts + τpc + τcp ] The parameters Φ3 to Φ7 , and Γ3 to Γ7 , are the same as in the previous subsection.
6 Compensating for Controller-to-Plant Time-Delay In the previous subsection, the estimate of the plant state, x ˆ[kts + τpc ], was computed based on the measured time-delay τpc . The resulting control signal u = −Kx[kts + τpc ] generated will arrive at the plant with a time-delay τcp , but unfortunately, at the time of computing the control signal, this controller-to-plant time-delay is unknown. However, assuming that we have the time stamps of the previous control signals applied to the plant, we can obtain an estimate of the next controller-to-plant time-delay.