On the control of distributed parameter systems using a multidimensional systems setting

Abstract

The unique characteristic of a repetitive process is a series of sweeps, termed passes, through a set of dynamics defined over a finite duration with resetting before the start of the each new one. On each pass an output, termed the pass profile is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. This leads to the possibility that the output, i.e. the sequence of pass profiles, will contain oscillations which increase in amplitude in the pass-to-pass direction. Such behavior cannot be controlled by standard linear systems approach and instead they must be treated as a multidimensional system, i.e. information propagation in more than one independent direction. Physical examples of such processes include long-wall coal cutting and metal rolling. In this paper, stability analysis and control systems design algorithms are developed for a model where a plane, or rectangle, of information is propagated in the passto-pass direction. The possible use of these in the control of distributed parameter systems is then described using a fourthorder wavefront equation. r

Robust control of distributed parameter mechanical systems using a multidimensional systems approach

“…The detailed derivation of this model can be found in [16]. Before proceeding, it is essential to verify if the model obtained is an acceptably accurate approximation to the original Bull.…”

“…In particular, we determine a relationship between ∆t and ∆x which guarantees convergence by applying von Neumann analysis (a standard technique in this general area). This yields the following requirement (for the details again see [16]). …”

“…This result can also be expressed in terms of a Linear Matrix Inequality (LMI), which provides a computational test for this property, [16].…”

“…A discrete linear repetitive process described by (4) and (5) is stable along the pass if there exist matrices V > 0 (defined in (16)) and G such that…”

Robust control with finite frequency specification for uncertain discrete linear repetitive processes

“…The importance of providing those results in the domain of LRPs is motivated by wide range of applications for modeling, analysis and synthesis with the LRPs framework (Azevedo-Perdicoulis and Jank 2012; Rabenstein and Steffen 2012;Wu et al 2011), especially for long-wall coal cutting, metal rolling (Rogers et al 2007), interconnected systems (Cichy 2008) and designing of iterative learning control (ILC) schemes (Bristow et al 2006;Rogers et al 2007). This is mostly why LRPs are the most investigated class of 2D systems.…”