Strong practical stability and stabilization of discrete linear repetitive processes

Abstract

This paper considers two-dimensional (2D) discrete linear systems recursive over the upper right quadrant described by well known state-space models. Included are discrete linear repetitive processes that evolve over subset of this quadrant. A stability theory exists for these processes based on a bounded-input bounded-output approach and there has also been work on the design of stabilizing control laws, elements of which have led to the assertion that this stability theory is too strong in many cases of applications interest. This paper develops so-called strong practical stability as an alternative in such cases. The analysis includes computationally efficient tests that lead directly to the design of stabilizing control laws, including the case when there is uncertainty associated with the process model. The results are illustrated by application to a linear model approximation of the dynamics of a metal rolling process.

Multidimensional control systems: case studies in design and evaluation

“…Strong practical stability of processes described by (15) holds by the analysis in [Dabkowski et al (2009) [Du and Xie (2002)] for 2D discrete linear systems.…”

New results on strong practical stability and stabilization of discrete linear repetitive processes

“…The case when k → ∞ and α → ∞ simultaneously cannot arise in physical applications. Using results in [6], strong practical stability of a process described by (6) and (7) requires that:…”

“…Previous work [6] developed strong practical stability for discrete linear repetitive processes where the boundedness property is not required to hold when k → ∞ and p → ∞, a combination of k and p that cannot arise in a physical application. For control law design, this stability property does not require frequency attenuation of the previous pass error over the complete spectrum.…”

Iterative Learning Control Based on Relaxed 2-D Systems Stability Criteria

“…Theorem 1 [6]: A discrete linear repetitive process described by (1) is strongly practically stable if and only if there exist matrices W 1 > 0, W 2 > 0, Q 1 > 0, Q 2 > 0, and nonsingular matrices S 1 and S 2 such that the following set of LMIs is feasible:…”

“…This brief gives new results where strong practical stability [6] for discrete linear repetitive processes is used, which leads to reduced complexity in design, mainly due to the need to solve lower order linear matrix inequalities (LMIs). The results of experimental application to gantry robot are given and discussed.…”