Abstract
Time-varying discrete-time linear systems may be temporarily uncontrollable and unreconstructable. This is vital knowledge to both control engineers and system scientists. Describing and detecting the temporal loss of controllability and reconstructability requires considering discrete-time systems with variable dimensions and the j-step, k-step Kalman decomposition. In this note for linear discrete-time systems with variable dimensions measures of temporal and one-step stabilizability and detectability are developed. These measures indicate to what extent the temporal loss of controllability and reconstructability may lead to temporal instability of the closed loop system when designing a static state or dynamic output feedback controller. The measures are calculated by solving specific linear quadratic cheap control problems.
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References
Athans, M.: The role and use of the Linear- Quadratic-Gaussian problem in control system design. IEEE Trans. Aut. Contr. 16, 529–552 (1971)
van Willigenburg, L.G., De Koning, W.L.: On the synthesis of time-varying LQG weights and noises along optimal control and state trajectories. Optimal Control Applications and Methods 27, 137–160 (2006)
Van Willigenburg, L.G., De Koning, W.L.: A Kalman decomposition to detect temporal linear system structure. In: Proceedings European Control Conference, Kos, Greece, July 2-7, Paper nr. 78, 6 p. (2007)
Van Willigenburg, L.G., De Koning, W.L.: Temporal linear system structure. IEEE Trans. Aut. Contr. 53(5), 1318–1323 (2008)
Van Willigenburg, L.G., De Koning, W.L.: Temporal and differential stabilizability and detectability of piecewise constant rank systems. Optimal Control Application & Methods, published online in Wiley Online Library (wileyonline library.com) (2011), doi:10.1002/oca.997
Jameson, A., O’Malley, R.E.: Cheap control of the time-invariant regulator. Appl. Math. & Optimization 1(4), 337–354 (1975)
Kokotovic, P.V., O’Malley, R.E., Sannuti, P.: Singular perturbations and order-reduction in control theory – an overview. Automatica 12, 123–132 (1976)
Levis, A.H., Schlueter, R.A., Athans, M.: On the behavior of optimal linear sampled-data regulators. International Journal of Control 13, 343–361 (1971)
Van Willigenburg, L.G., De Koning, W.L.: The digital optimal regulator and tracker for stochastic time-varying systems. International Journal of Systems Science 12, 2309–2322 (1992)
Van Willigenburg, L.G.: Digital optimal control and LQG compensation of asynchronous and aperiodically sampled nonlinear systems. In: Proceedings 3rd European Control Conference, Rome, Italy, vol. 1, pp. 496–500 (September 1995)
van Willigenburg, L.G., De Koning, W.L.: Temporal linear system structure: The discrete-time case. In: Proceedings of the ECC 2009, Budapest, August 23-26, pp. 225–230 (2009)
Kalman, R.E., Bertram, J.E.: Control system design via the “Second Method” of Lyapunov, I Continuous-time systems. Transactions of the ASME, Journal of Basic Engineering, 371–393 (June 1960)
Amato, F., Ariola, M., Carbone, M., Cosentiono, C.: Finite-Time Control of Linear Systems: A Survey. In: Current Trends in Nonlinear Systems and Control. Systems and Control: Foundations & Applications, Part II, pp. 195–213 (2006)
Amato, F., Ambrosino, R., Ariola, M., Cosentino, C.: Finite-time stability of linear time-varying systems with jumps. Automatica 45, 1354–1358 (2009)
Gohberg, I., Kaashoek, M.A., Lerer, L.: Minimality and realization of discrete time-varying systems. Operator Theory: Advances and Applications 56, 261–296 (1992)
Van Willigenburg, L.G., De Koning, W.L.: Minimal and non-minimal optimal fixed-order compensators for time-varying discrete-time systems. Automatica 38, 157–165 (2002)
Sandberg, H., Rantzer, A.: Balanced truncation of linear time-varying systems. IEEE Trans. Aut. Contr. 49(2), 217–229 (2004)
Van Willigenburg, L.G., De Koning, W.L.: Linear systems theory revisited. Automatica 44, 1686–1696 (2008)
Boley, D.: Computing the Kalman decomposition: An optimal method. IEEE Trans. Aut. Contr. 29(1), 51–53 (1984)
van Willigenburg, L.G., De Koning, W.L.: Compensatability and optimal compensation of systems with white parameters in the delta domain. International Journal of Control 83(12), 2546–2563 (2010)
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Van Willigenburg, L.G., De Koning, W.L. (2013). Temporal and One-Step Stabilizability and Detectability of Time-Varying Discrete-Time Linear Systems. In: Hömberg, D., Tröltzsch, F. (eds) System Modeling and Optimization. CSMO 2011. IFIP Advances in Information and Communication Technology, vol 391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36062-6_31
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DOI: https://doi.org/10.1007/978-3-642-36062-6_31
Publisher Name: Springer, Berlin, Heidelberg
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