Practical-BIBO stability of n-dimensional discrete systems

Galkowskiy

IEEE Trans. Contr. Syst. Technol.

et al. 2013

Abstract-This brief develops a new algorithm for the design of iterative learning control law algorithms in a 2-D systems setting. This algorithm enables control law design for error convergence and performance, and is actuated by process output information only. Results are also given from the experimental application to a gantry robot.Index Terms-2-D systems design, iterative learning control, repetitive processes.

Section: Stability and Convergence Analysismentioning

confidence: 99%

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

Galkowskiy

IEEE Trans. Contr. Syst. Technol.

et al. 2013

“…To explain the term strong practical stability, first note that for 2D discrete linear systems practical stability [6] was introduced due to a recognition that the BIBO stability theory for such systems was too strong for some applications, and practical stability replaced this by the requirement the response in each direction of information propagation is stable. Practical stability can also be defined for differential linear repetitive processes described by (1) and would require conditions [a] and [b] above to hold which, as the simple example shows, is too weak in some cases.…”

Section: Remarkmentioning

confidence: 99%

Strong practical stability and stabilization of differential linear repetitive processes

Gałkowski

Systems & Control Letters

et al. 2010

a b s t r a c tDifferential linear repetitive processes evolve over a subset of the upper-right quadrant of the 2D plane where the unique feature is a series of sweeps or passes through a set of dynamics governed by the solution of a linear matrix differential equation over a finite duration t ∈ [0, α] where α is termed the pass length or duration. 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. The result can be oscillations in the pass-to-pass direction that cannot be controlled by direct application of standard, or 1D linear systems theory. The existing stability theory demands a bounded-input bounded-output property uniformly, which in the case of the along-the-pass dynamics means for t ∈ [0, ∞] The pass length is always finite, however, and hence this stability theory could well be too strong in many cases and, in particular, impose very strong conditions in terms of control law design. This paper develops an alternative in such cases by relaxing the requirement for the bounded-input bounded-output property to hold when k → ∞ and t → ∞ simultaneously, provides an explanation of the implications of this in the frequency domain, and then develops control law design algorithms.

“…As already noted in this paper, another way to consider stability of 2D systems, and hence discrete linear repetitive processes of the form considered here, is to use the weaker concept of practical stability (Agathoklis and Bruton 1983;Xu et al 1994Xu et al , 1997. Applying the resulting conditions to the 2D linear systems interpretation of processes described by (1) and (2) gives the following result.…”

Section: Notementioning

confidence: 99%

“…This raised the question of whether or not there is another useful definition of this property and led to so-called practical BIBO stability for nD linear systems-see, for example, Agathoklis and Bruton (1983); Xu et al (1994Xu et al ( , 1997. The stability equivalence (Boland and Owens 1980) means that the practical stability analysis can also be applied to discrete linear repetitive processes (obviously the same sub-class for which this equivalence holds).…”

mentioning

confidence: 99%

Strong practical stability and stabilization of discrete linear repetitive processes

Gałkowski

Multidim Syst Sign Process

et al. 2009

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.