LMI based stability analysis and robust controller design for discrete linear repetitive processes

Abstract

SUMMARYDiscrete linear repetitive processes are a distinct class of 2D linear systems with applications in areas ranging from long-wall coal cutting through to iterative learning control schemes. The main feature which makes them distinct from other classes of 2D linear systems is that information propagation in one of the two independent directions only occurs over a finite duration. This, in turn, means that a distinct systems theory must be developed for them. In this paper, the major new development is that an LMI based reformulation of the stability conditions can used to enable the design of a family of control laws which have a well defined physical basis. It is also noted that this setting can be used to investigate robustness aspects.

Robust Control of Discrete Linear Repetitive Processes with Parameter Varying Uncertainty

“…In addition to Theorem 1.2 of the previous section, the design of control laws for discrete linear repetitive processes has been the subject of much research effortsee, for example, [5,6,7,14]. Here, we continue the development of this general area by establishing new results related to the practical case where there is possibly large uncertainty associated with the process (state-space model) description.…”

“…0: Numerous sets of necessary and sufficient conditions for stability along the pass of (9.2) and (9.3) are known but here it is the following result which is the starting point. Theorem 1.1 [6]. A discrete linear repetitive process described by (9.2) and (9.3) is stable along the pass if there exist matrices W 1 1 0 and W 2 1 0 such that…”

“…In effect, this control law is composed of a weighted sum of current pass state feedback and feedforward of the previous pass profile. The LMI of (9.5) extends in a natural manner to the design of (9.10) for stability along the pass but here we will use the approach based on [2,9,15] and first adopted for repetitive processes in [6]. This will prove to be of particular use in the analysis of the case when there is uncertainty associated with the process state-space model.…”

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

“…Better computational results can be obtained based on [14] and first adopted for repetitive processes in [15].…”

Decoupling and iterative approaches to the control of discrete linear repetitive processes

“…In common with a large range of other areas in systems theory, recent years has seen the emergence of Linear Matrix Inequality (LMI) based techniques in the analysis of very important sub-classes of linear repetitive processes (see, e.g., Galkowski, Rogers, Xu, Lam, & Owens, 2002;Galkowski, Paszke, Sulikowski, Rogers, & Owens, 2003). This has led to considerable success, especially in areas such as the structure and design of control laws and stability in the presence physically relevant types of uncertainty in the matrices of the defining state-space model, which have proved very difficult/impossible to handle using other analysis tools.…”