Second-order counterexamples to the discrete-time Kalman conjecture. (English) Zbl 1331.93168
Summary: The Kalman conjecture is known to be true for third-order continuous-time systems. We show that it is false in general for second-order discrete-time systems by construction of counterexamples with stable periodic solutions. We discuss a class of second-order discrete-time systems for which it is true provided the nonlinearity is odd, but false in general. This has strong implications for the analysis of saturated systems.
MSC:
| 93D09 | Robust stability |
| 93C55 | Discrete-time control/observation systems |
| 93C10 | Nonlinear systems in control theory |
References:
| [1] | Ahmad, N. S.; Carrasco, J.; Heath, W. P., LMI searches for discrete-time Zames-Falb multipliers, (Proceedings of the IEEE Conference on Decision and Control (2013)), 5258-5263 |
| [2] | Ahmad, N. S.; Carrasco, J.; Heath, W. P., A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities, IEEE Transactions on Automatic Control, 60, 6, 1692-1697 (2015) · Zbl 1360.93527 |
| [3] | Ahmad, N. S.; Heath, W. P.; Li, G., LMI-based stability criteria for discrete-time Lur’e systems with monotonic, sector- and slope-restricted nonlinearities, IEEE Transactions on Automatic Control, 58, 2, 459-465 (2013) · Zbl 1369.93469 |
| [4] | Aizerman, M. A.; Gantmacher, F. R., Absolute stability of regular systems (1964), Holden-Day · Zbl 0123.28401 |
| [5] | Altshuller, D., Frequency domain criteria for absolute stability: a delay-integral-quadratic constraints approach (2013), Springer · Zbl 1401.93001 |
| [6] | Barabanov, N. E., On the Kalman problem, Siberian Mathematical Journal, 29, 333-341 (1988) · Zbl 0713.93044 |
| [7] | Bragin, V.; Vagaitsev, V.; Kuznetsov, N.; Leonov, G., Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits, Journal of Computer and Systems Sciences International, 50, 4, 511-543 (2011) · Zbl 1266.93072 |
| [8] | Brogliato, B.; Maschke, B.; Lozano, R.; Egeland, O., Dissipative systems analysis and control: theory and applications (2006), Springer |
| [9] | Carrasco, J.; Heath, W. P.; de la Sen, M., Second-order counterexample to the discrete-time Kalman conjecture, (Proceedings of the European control conference, ECC15 (2015)), 975-979 |
| [10] | Carrasco, J.; Heath, W. P.; Li, G.; Lanzon, A., Comments on “on the existence of stable, causal multipliers for systems with slope-restricted nonlinearities”, IEEE Transactions on Automatic Control, 57, 9, 2422-2428 (2012) · Zbl 1369.93478 |
| [11] | Carrasco, J.; Maya-Gonzalez, M.; Lanzon, A.; Heath, W. P., LMI searches for anticausal and noncausal rational Zames-Falb multipliers, Systems & Control Letters, 70, 17-22 (2014) · Zbl 1290.93136 |
| [12] | Carrasco, J.; Turner, M. C.; Heath, W. P., Zames-Falb multipliers for absolute stability: from O’Shea’s contribution to convex searches, (Proceedings of the European control conference, ECC15 (2015)), 1261-1278 |
| [13] | Chang, M.; Mancera, R.; Safonov, M., Computation of Zames-Falb multipliers revisited, IEEE Transactions on Automatic Control, 57, 4, 1024-1029 (2012) · Zbl 1369.93470 |
| [14] | Chen, X.; Wen, J., Robustness analysis for linear time invariantsystems with structured incrementally sector bounded feedback non-linearities, Applied Mathematics and Computer Science, 6, 623-648 (1996) · Zbl 0870.93011 |
| [15] | Fitts, R., Two counterexamples to Aizerman’s conjecture, IEEE Transactions on Automatic Control, 11, 3, 553-556 (1966) |
| [16] | Gonzaga, C. A.; Jungers, M.; Daafouz, J., Stability analysis of discrete-time Lur’e systems, Automatica, 48, 9, 2277-2283 (2012) · Zbl 1257.93060 |
| [17] | Haddad, W. M.; Bernstein, D. S., Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle and Popov theorems and their application to robust stability. Part II: Discrete time theory, International Journal of Robust and Nonlinear Control, 4, 2, 249-265 (1994) · Zbl 0806.93045 |
| [18] | Haddad, W. M.; Chellaboina, V., Nonlinear dynamical systems and control (2008), Princeton University Press · Zbl 1152.93456 |
| [19] | Heath, W. P.; Carrasco, J., Global asymptotic stability for a class of discrete-time systems, (Proceedings of the European control conference, ECC15 (2015)), 963-968 |
| [20] | Heath, W. P.; Wills, A. G., Zames-Falb multipliers for quadratic programming, IEEE Transactions on Automatic Control, 52, 10, 1948-1951 (2007) · Zbl 1366.90155 |
| [21] | Hu, T.; Lin, Z., A complete stability analysis of planar discrete-time linear systems under saturation, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48, 6, 710-725 (2001) · Zbl 1104.93315 |
| [22] | Hu, T.; Lin, Z.; Chen, B. M., Analysis and design for discrete-time linear systems subject to actuator saturation, Systems & Control Letters, 45, 2, 97-112 (2002) · Zbl 0987.93027 |
| [23] | Kalman, R. E., Physical and mathematical mechanisms of instability in nonlinear automatic control systems, Transactions of the ASME, 79, 3, 553-566 (1957) |
| [24] | Kapila, V.; Haddad, W. M., A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach, IEEE Transactions on Automatic Control, 41, 1, 149-152 (1996) · Zbl 0842.93058 |
| [25] | LaSalle, J. P., The stability of dynamical systems (1976), SIAM · Zbl 0364.93002 |
| [26] | Leonov, G. A.; Kuznetsov, N. V., Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems, Doklady Mathematics, 84, 1, 475-481 (2011) · Zbl 1247.34063 |
| [27] | Leonov, G. A.; Kuznetsov, N. V., Hidden attractors in dynamical systems. from hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, International Journal of Bifurcation and Chaos, 23, 1, 1-69 (2013) · Zbl 1270.34003 |
| [28] | Leonov, G.; Kuznetsov, N.; Bragin, V., On problems of Aizerman and Kalman, Vestnik St. Petersburg University: Mathematics, 43, 148-162 (2010) · Zbl 1251.93110 |
| [29] | Megretski, A.; Rantzer, A., System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control, 42, 6, 819-830 (1997) · Zbl 0881.93062 |
| [30] | Park, P.; Kim, S. W., A revisited Tsypkin criterion for discrete-time nonlinear Lur’e systems with monotonic sector-restrictions, Automatica, 34, 11, 1417-1420 (1998) · Zbl 1040.93531 |
| [31] | Park, B. Y.; Park, P.; Kwon, N. K., An improved stability criterion for discrete-time Lur’e systems with sector- and slope-restrictions, Automatica, 51, 1, 255-258 (2015) · Zbl 1417.93241 |
| [32] | Safonov, M. G.; Wyetzner, G., Computer-aided stability analysis renders Popov criterion obsolete, Transactions on Automatic Control, 32, 12, 1128-1131 (1987) · Zbl 0636.93060 |
| [33] | Tarbouriech, S.; Garia, G.; Gomes da Silva, J. M.; Qieinnec, I., Stability and stabilization of linear systems with saturating actuators (2011), Springer · Zbl 1279.93004 |
| [34] | Tsypkin, Y. Z., On the stability in the large of nonlinear sampled-data systems, Doklady Akademii Nauk SSSR, 145, 52-55 (1962) |
| [35] | Turner, M. C.; Kerr, M. L.; Postlethwaite, I., On the existence of stable, causal multipliers for systems with slope-restricted nonlinearities, IEEE Transactions on Automatic Control, 54, 11, 2697-2702 (2009) · Zbl 1367.93543 |
| [36] | Vidyasagar, M., Nonlinear systems analysis (1993), Prentice-Hall International Editions, reprinted SIAM 2002 · Zbl 0900.93132 |
| [37] | Wang, S.; Heath, W. P.; Carrasco, J., A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers, (Proceedings of the IEEE conference on decision and control (2014)) |
| [38] | Willems, J.; Brockett, R., Some new rearrangement inequalities having application in stability analysis, IEEE Transactions on Automatic Control, 13, 5, 539-549 (1968) |
| [39] | Yang, T.; Stoorvogel, A. A.; Saberi, A., Dynamic behavior of the discrete-time double integrator with saturated locally stabilizing linear state feedback laws, International Journal of Robust and Nonlinear Control, 23, 17, 1899-1931 (2013) · Zbl 1278.93208 |
| [40] | Zaccarian, L.; Teel, A. R., Modern anti-windup synthesis (2011), Princeton · Zbl 1293.93434 |
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.
