Estimation of linear positive systems with unknown time-varying delays

Delay systems have attracted a considerable amount of interest δðtnÞ for t≥t, which in practice is unrealistic. Conceptually, this in the control theory community in recent years, due to important methodological and practical issues involved in their analysis and design. New and emergent application areas, such as control of networked systems, are inherently characterized by the presence of time delays [7]. Stability analysis, control design, as well as estimation problems for both linear and nonlinear systems with delays in the input, output or state equations have been studied in an impressive number of papers [8]. Due to the inherent complexity of time-delay systems, it is well known that there is still much to be done even in the relatively simple context of linear timeinvariant systems. An important issue in this area is to achieve results that are general enough with respect to the nature of the delay. The trend of research results with respect to the kind of delay which has moved from the case of a single constant delay to multiple constant delays, time varying continuous delays, timevarying bounded (and non-necessarily continuous) delays. It is likely that the case of unknown delays is the next important step forward, and for this reason, the paper [2] of Rami, Jordan and Schönlein is particularly relevant, since it is one of the first attempts to provide a methodological framework in which this difficult problem can be casted and solved. I will briefly discuss three issues, namely (i) the importance of the contribution, (ii) its limitations and (iii) the points that in my opinion can be further developed. Why unknown delays are important. Consider the problem of controlling a linear system affected by time varying delays in the actuator or in the measurement process. It is reasonable to assume that in practice the delay function δðtÞ on the output is bounded and known at time t, since this can be achieved through the timestamps associated to the measurement. If the delay is due to the communication on a shared network, the delay function may be not continuous, but it is still known and bounded. Interestingly, however, when the delay affects the input the knowledge of δðtÞ at time t is not sufficient to generate a feedback with a delay dependent gain, since the controller would not know which delay affects the input signal generated at time t. This delay is δðtnÞ, with t 1⁄4 tn−δðtnÞ, and therefore t≥t (notice that t might also be not unique). Thus, in the case of delayed input it is necessary to know