A graph-oriented approach to address generically flat outputs in structured LTI discrete-time systems. (English) Zbl 1494.93069
Summary: This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.
MSC:
| 93C55 | Discrete-time control/observation systems |
| 93C05 | Linear systems in control theory |
| 05C90 | Applications of graph theory |
References:
| [1] | Boukhobza, T., Sensor location for discrete mode observability of switching structured linear systems with unknown input, Automatica, 48, 7, 1262-1272 (2012) · Zbl 1246.93021 |
| [2] | Boukhobza, T.; Hamelin, F., Observability analysis for structured bilinear systems: A graph-theoretic approach, Automatica, 43, 11, 1968-1974 (2007) · Zbl 1129.93331 |
| [3] | Boukhobza, T., & Millérioux, G. (2016). Graphical characterization of the set of all flat outputs for structured linear discrete-time systems. In 6th symposium on system structure and control. |
| [4] | Clark, A.; Alomair, B.; Bushnell, L.; Poovendran, R., Input selection for performance and controllability of structured linear descriptor systems, SIAM Journal on Control and Optimization, 55, 1, 457-485 (2017) |
| [5] | Commault, C.; Dion, J.; van der Woude, J. W., Characterization of generic properties of linear structured systems for efficient computations, Kybernetika, 38, 503-520 (2002) · Zbl 1265.93120 |
| [6] | Daafouz, J., Fliess, M., & Millérioux, G. (2006). Une approche intrinsèque des observateurs linéaires à entrées inconnues. In Proc. of the conférence internationale francophone d’automatique. |
| [7] | Dakil, M.; Simon, C.; Boukhobza, T., Generic methodology for the probabilistic reliability assessment of some structural properties: A graph theoretical approach, International Journal of Systems Science, 46, 10, 1825-1838 (2015) · Zbl 1332.93331 |
| [8] | Dion, J.-M.; Commault, C.; van der Woude, J. W., Generic properties and control of linear structured systems: A survey, Automatica, 39, 7, 1125-1144 (2003) · Zbl 1023.93002 |
| [9] | Diwold, J.; Kolar, B.; Schöberl, M., A trajectory-based approach to discrete-time flatness, IEEE Control Systems Letters, 6, 289-294 (2022) |
| [10] | Edmonds, J.; Karp, R., Theoretical improvements in algorithmic efficiency for network flow problems, Journal of the ACM, 19, 2, 248-264 (1972) · Zbl 0318.90024 |
| [11] | Fliess, M., Reversible linear and nonlinear discrete-time dynamics, IEEE Transactions on Automatic Control, 37, 8, 1144-1153 (1992) · Zbl 0764.93058 |
| [12] | Fliess, M.; Levine, J.; Martin, P.; Rouchon, P., Flatness and defect of non-linear systems: introductory theory and examples, International Journal of Control, 61, 6, 1327-1361 (1995) · Zbl 0838.93022 |
| [13] | Fliess, M.; Marquez, R., Toward a module theoretic approach to discrete-time linear predictive control, International Journal of Control, 73, 606-623 (2000) · Zbl 1006.93508 |
| [14] | Gracy, S., Milosevic, J., & Sandberg, H. (2020). Actuator security index for structured systems. In Proceedings of American control conference. · Zbl 1478.93242 |
| [15] | Guillot, P.; Millérioux, G., Flatness and submersivity of discrete-time dynamical systems, IEEE Control Systems Letters, 4, 2, 337-342 (2020) |
| [16] | Kaldmäe, A.; Kotta, U., On flatness of discrete-time nonlinear systems. in proceedings of the 9th IFAC symposium on nonlinear control systems. Toulouse, France. Experiments, IEEE Transactions on Signal Processing, 63, 13 (2013) |
| [17] | Kawano, Y.; Cao, M., Structural accessibility and its applications to complex networks governed by nonlinear balance equations, IEEE Transactions on Automatic Control, 64, 11, 4607-4614 (2019) · Zbl 1482.93090 |
| [18] | Kolar, B., Kaldmäe, A., Schöberl, M., Kotta, U., & Schlacher, K. (2016). Construction of flat outputs of nonlinear discrete-time systems in a geometric and an algebraic framework. In Proceedings of the 10th non-linear control systems. |
| [19] | Kolar, B., Schöberl, M., & Schlacher, K. (2016). A decomposition procedure for the construction of flat outputs of discrete-time nonlinear control systems. In Proceedings of the 22nd international symposium on mathematical theory of networks and systems. |
| [20] | Liu, X.; Linqiang, P., Identifying driver nodes in the human signalling network using structural controllability analysis, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 12, 2, 467-472 (2015) |
| [21] | Millérioux, G.; Daafouz, J., Flatness of switched linear discrete-time systems, IEEE Transactions on Automatic Control, 54, 3, 615-619 (2009) · Zbl 1367.93268 |
| [22] | Parriaux, J.; Millérioux, G., Nilpotent semigroups for the characterization of flat outputs of switched linear and LPV discrete-time systems, Systems & Control Letters, 62, 8, 679-685 (2013) · Zbl 1279.93036 |
| [23] | Ramos, G.; Aguiar, A. P.; Pequito, S. D., Structural systems theory: an overview of the last 15 years (2020), ArXiv abs/2008.11223 |
| [24] | Reinschke, K. J., Multivariable control. a graph theoretic approach (1988), Springer-Verlag: Springer-Verlag New York, U.S.A · Zbl 0682.93006 |
| [25] | Sato, K. (2012). On an algorithm for checking whether or not a nonlinear discrete-time system is difference flat. In Proceedings of 20th international symposium on mathematical theory of networks and systems. |
| [26] | Shoukry, Y., Nuzzo, P., Bezzo, N., Sangiovanni-Vincentelli, A. L., Seshia, S. A., & Tabuada, P. (2015). Secure state reconstruction in differentially flat systems under sensor attacks using satisfiability modulo theory solving. In Proceedings of the 54th IEEE conference on decision and control. · Zbl 1390.93532 |
| [27] | Sira-Ramirez, H.; Agrawal, S. K., Differentially flat systems (2004), Marcel Dekker: Marcel Dekker New York. USA · Zbl 1126.93003 |
| [28] | van der Woude, J. W., On the structure at infinity of a structured system, Linear Algebra and Its Applications, 148, 145-169 (1991) · Zbl 0724.93019 |
| [29] | van der Woude, J. W., The generic number of invariant zeros of a structured linear system, SIAM Journal on Control and Optimization, 38, 1, 1-21 (2000) · Zbl 0952.93056 |
| [30] | van der Woude, C.; Dion, J. M., Zero orders and dimensions of some invariant subspaces in linear structured systems, Mathematics of Control, Signals, and Systems, 16, 2-3, 225-237 (2003) · Zbl 1029.93033 |
| [31] | Yamada, T., A network flow algorithm to find an elementary I/O, matching, Networks, 18, 105-109 (1988) · Zbl 0641.90039 |
| [32] | Yong, S. Z., Paden, B., & Frazzoli, E. (2015). Computational methods for MIMO flat linear systems: Flat output characterization, test and tracking control. In Proceedings of the American control conference. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
