Linear Algebra for Control Theory pp 103-116 | Cite as

# The Block form of Linear Systems over Commutative Rings with Applications to Control

## Abstract

This paper deals with a new approach to the study of linear time-invariant discrete-time systems whose coefficients belong to an arbitrary commutative ring. Such systems arise in the study of integer systems, systems depending on parameters, and multi-dimensional systems. The key idea is to consider the representation of systems over a ring in terms of a block-input/block-output form. By time-compressing the block representation, new results are derived on assignability by state feedback control including the construction of deadbeat controllers. A new type of state observer is then considered based on a block-output form for the update term in the state estimate. The results on state observers are combined with the time-compression approach to state feedback control to yield a new type of input/output regulator. In the last section of the paper the results are applied to the problem of state and output tracking of set points.

## Key words

Systems over rings reachability observability feedback control observers regulators tracking## AMS(MOS) subject classifications

93B05 93B07 93B25 93C05 93C45 93C55 93D15## Preview

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## References

- [1]E.W. Kamen,
*Linear systems over rings: from R.E. Kaiman to the present*, in Mathematical System Theory-The Influence of R.E. Kaiman, A.C. Antoulas (Ed.), Springer-Verlag, Berlin (1991), pp. 311–324.Google Scholar - [2]P. Albertos,
*Block multirate input-output model for sampled control systems*, IEEE Trans, on Automatic Control, 35 (1990), pp. 1085–1088.MathSciNetzbMATHCrossRefGoogle Scholar - [3]P. Albertos and R. Ortega,
*On generalized predictive control: two alternative formulations*, Automatica, 25 (1989), pp. 753–755.zbMATHCrossRefGoogle Scholar - [4]R. Lozano,
*Robust adaptive regulation without persistent excitation*, IEEE Trans. on Automatic Control, 34 (1989), pp. 1260–1267.zbMATHCrossRefGoogle Scholar - [5]E.W. Kamen,
*Study of linear time-varying discrete-time systems in terms of time-compressed models*. Proceedings of the 31st IEEE Decision and Control Conference, Tucson, AZ (1992), pp. 3070–3075.Google Scholar - [6]E.W. Kamen,
*Regulation of time-varying discrete-time systems based on a time-compressed formulation*, Proceedings of the International Symposium on Implicit and Nonlinear Systems, Arlington, TX (1992), pp. 267–274.Google Scholar - [7]J.W. Brewer, J.W. Bunce, and F.S. Vanvleck,
*Linear systems over commutative rings*, Marcel Dekker, New York, 1986.zbMATHGoogle Scholar