State-space parametrization of all stabilizing dead-time controllers

Abstract

This paper considers the stabilization problem for systems with a single delay h in the feedback loop. The state-space parametrizations of all stabilizing regulators are derived. These parametrizations have simple structures and clear interpretations. In particular, it is shown that every stabilizing controller consists of a delayed state observer, an h time units ahead predictor and a stabilizing state feedback. Applications of the proposed parametrization to the H2 control and robust stabilization are discussed.

H control of system with I/O delay: a review of some problem-oriented methods

“…The next step is then to extract all Q a which bring the LFT in the form e −τ s K τ for a proper K τ . This can easily be done by the loop-shifting arguments of [44] yielding the following characterization of all admissible Q a : Q a = ∆ 22 + e −τ s Q b , where Q b ∈ H ∞ is arbitrary and ∆ 22 is the FIR truncation of the (2, 2) subblock of G −1 . Now, taking into account the norm constraint on Q a one can see that the original (four-block) problem is reduced to the (one-block) problem of the characterization of all Q b ∈ H ∞ such that:…”

“…Some other possible choices are discussed in [40]. Moreover, as shown in [44] the set of all stabilizing dead-time controllers can be parameterized in the DTC form shown in Fig. 2.…”

“…Moreover, if the observability or/and controllability of (C 1 , A, B 1 ) are relaxed, then small enough delay might not affect the achievable performance even when the spectral radius condition fails first. This, for instance, occurs in the case when A is Hurwitz and P 11 = 0, see [44].…”

Stabilization of MISO fractional systems with delays

“…For the particular class of MIMO (integer-order) systems with I/O delays, the problem of parametrization of stabilizing controllers was solved in [8,9]. The idea was to reduce the problem to an equivalent finite-dimensional stabilization problem by involving an unstable finitedimensional system and a stable infinite-dimensional system (FIR filter).…”

“…Our strategy here is to work directly on the Bézout identity in order to get explicit expressions of Bézout factors in terms of the matrix transfer function. Such explicit expressions could not be easily derived in [8,9,10] even in the case of standard delay systems. We hope that the explicit form will facilitate the use of these factors in controllers design while the use of the frequency domain representation of the systems agrees well with the modeling practice of fractional systems [17].…”