Sensitivity integrals and transformation techniques: a new perspective

Abstract

In this note we provide alternative proofs for the classical Bode and Poisson type sensitivity integrals and their extensions for both continuous-time and discrete-time systems. Our derivation uses the wellknown properties of Laplace and Z-transformations. This derivation helps establish a connection between Bode and Poisson type sensitivity integrals and Laplace and Z-transforms, hence providing an alternative perspective in interpreting these fundamental integral results.

Performance Recovery in Digital Implementation of Analogue Systems

“…Therefore, the open-loop transfer function G d M is also strictly proper, and its unstable poles are given uniquely by G d . According to Theorem 2 of [36] (resp., Theorem 3.3 of [8]), the Bode integral of the sensitivity function from −π to π is around 40.8407, which is much bigger than that of the sensitivity function G gbt F . Due to space limitation, we omit the calculation of Bode integrals for complementary functions of G d M and G gbt F .…”

“…Hence, the resulting open-loop transfer function G gbt F is strictly proper too. In light of this, according to Theorem 2 of [36] (resp., Theorem 3.3 of [8]), the Bode integral of the sensitivity function from −π to π is 2π · (log(p gbt 1 )) ≈ 2π · 1.3350 ≈ 8.3881. As to G d , it is not easy to find directly a strictly proper stable controller M that ensures stability of the closed-loop system.…”

Nonlinear control designs for hard disk drive servos

“…Previous study showed that there are various limitations and tradeoffs associated with linear control systems ( [53], [110] and [141]). Typically, Bode gain-phase relation [9], Bode integral theorem [17], and time-domain constraint [113] are particularly restricting the performance of an HDD servo system. Much of the motivation for the research, which is summarized in this thesis comes from the application of nonlinear control theory to the head positioning servomechanism in HDD mechatronics, aims to develop nonlinear control techniques which are potential of breaking the performance limitations in linear control systems.…”

Optimal reset control and disturbance compensation for advanced hard disk drives

“…Typically, Bode gainphase relation, Bode's integral theorem [58] and time-domain constraint. In recent years, some new nonlinear control algorithms [59,60] had been introduced and developed that aims to improve HDD servo performance by means of breaking the performance limitations in linear control systems.…”