Output feedback stabilization of linear systems with infinite distributed input and output delays

Output feedback stabilization of linear time‐varying systems with infinite distributed input delays

“…Remark It should be pointed out that the delay kernel $$$ {B}_0\left(t,\eta \right) $$$ needs to be known in this work. This is similar to many existing works on distributed delays including bounded distributed delays 18 and infinite distributed delays 28‐32 . It should also be noted that the proposed observer is different from commonly used observers, that is, this observer does not contain the part of the initial control signal, namely, $$$ {\int}_t^{+\infty }{B}_0\left(t,\eta \right)u\left(t-\eta \right) d\eta $$$.…”

“…In contrast to Reference 32 where state feedback control of a special class of LTV systems of the converging system matrix with infinite distributed input delays is considered, this work considers output feedback control of more general periodic LTV systems with infinite distributed input delays. Moreover, the newly proposed observer is designed so that it does not contain the part of the initial control signal contrary to commonly designed observers. Unlike References 28‐32 where a restrictive assumption is implicitly made on the initial control signal during $$$ t\in \left(-\infty, 0\right) $$$, namely, the initial control signal is required to have the same form as the proposed controller, this work only requires a more general and less restrictive condition on the initial control signal by regarding the part of the initial control signal as perturbations, which is presented in Assumption 4. The obtained results in this work include some existing works on bounded distributed delays as special cases, for instance, References 40 and 18. To our best knowledge, the output feedback stabilization problem of LTV systems with infinite distributed input delays is considered for the first time. It is observed that the closed‐loop control system is globally exponentially stable (GES) as long as all the characteristic multipliers of the system matrix are on the unit circle.…”

“…In fact, many existing works on LTV systems with bounded delays also require this constraint, see, for example, References 41 and 42. This constraint reduces to $$$ \lambda (A)\subset {\mathbb{C}}^0 $$$ when the concerned system reduces to an LTI system, as required in References 28‐32 for infinite distributed input delays. Assumption 2 is a standard assumption on output feedback control.…”

“…Since systems with infinite distributed input delays always contain a part of the initial control function, their solutions are very sensitive to the initial condition. As a result, those works [28][29][30][31][32] all rely on this restrictive assumption on the initial control signal to obtain the stabilization results. (iii) Other general challenges in handling infinite delays also exist in this work, for instance, no delayed bound can be used to scale the integrals.…”

“…Unlike References 28‐32 where a restrictive assumption is implicitly made on the initial control signal during $$$ t\in \left(-\infty, 0\right) $$$, namely, the initial control signal is required to have the same form as the proposed controller, this work only requires a more general and less restrictive condition on the initial control signal by regarding the part of the initial control signal as perturbations, which is presented in Assumption 4.…”