On the necessity and sufficiency of discrete-time O’Shea-Zames-Falb multipliers. (English) Zbl 1519.93170
Summary: This paper considers the robust stability of a discrete-time Lurye system consisting of the feedback interconnection between a linear system and a bounded and monotone nonlinearity. It has been conjectured that the existence of a suitable linear time-invariant (LTI) O’Shea-Zames-Falb multiplier is not only sufficient but also necessary. Roughly speaking, a successful proof of the conjecture would require: (a) a conic parameterisation of a set of multipliers that describes exactly the set of nonlinearities, (b) a lossless S-procedure to show that the non-existence of a multiplier implies that the Lurye system is not uniformly robustly stable over the set of nonlinearities, and (c) the existence of a multiplier in the set of multipliers used in (a) implies the existence of an LTI multiplier. We investigate these three steps, showing the current bottlenecks for proving this conjecture. In addition, we provide an extension of the class of multipliers which may be used to disprove the conjecture.
MSC:
| 93D09 | Robust stability |
| 93C55 | Discrete-time control/observation systems |
| 93C10 | Nonlinear systems in control theory |
References:
| [1] | Carrasco, J.; Heath, W. P.; Lanzon, A., Equivalence between classes of multipliers for slope-restricted nonlinearities, Automatica, 49, 6, 1732-1740 (2013) · Zbl 1360.93519 |
| [2] | Carrasco, J.; Heath, W. P.; Zhang, J.; Ahmad, N. S.; Wang, S., Convex searches for discrete-time Zames-Falb multipliers, IEEE Transactions on Automatic Control, 65, 11, 4538-4553 (2020) · Zbl 1536.93387 |
| [3] | Carrasco, J.; Turner, M. C.; Heath, W. P., Zames-Falb multipliers for absolute stability: From O’Shea’s contribution to convex searches, European Journal of Control, 28, 1-19 (2016) · Zbl 1336.93120 |
| [4] | Clarke, F., Functional analysis, calculus of variations and optimal control, vol. 264 (2013), Springer · Zbl 1277.49001 |
| [5] | Curtain, R. F.; Zwart, H., An introduction to infinite-dimensional linear systems theory, vol. 21 (2012), Springer Science & Business Media |
| [6] | Dahleh, M. A.; Diaz-Bobillo, I. J., Control of uncertain systems: A linear programming approach (1994), Prentice-Hall, Inc. |
| [7] | Desoer, C. A.; Vidyasagar, M., Feedback systems: Input-output properties (2009), SIAM · Zbl 0327.93009 |
| [8] | Fetzer, M.; Scherer, C. W., Absolute stability analysis of discrete time feedback interconnections, IFAC-PapersOnLine, 50, 1, 8447-8453 (2017) |
| [9] | Freeman, R. A. (2018). Noncausal Zames-Falb Multipliers for Tighter Estimates of Exponential Convergence Rates. In 2018 annual American control conference (pp. 2984-2989). |
| [10] | Georgiou, T. T.; Smith, M. C., Robustness analysis of nonlinear feedback systems: An input-output approach, IEEE Transactions on Automatic Control, 42, 9, 1200-1221 (1997) · Zbl 0889.93043 |
| [11] | Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1952), Cambridge University Press · Zbl 0047.05302 |
| [12] | Heath, W. P.; Carrasco, J.; de la Sen, M., Second-order counterexamples to the discrete-time Kalman conjecture, Automatica, 60, 140-144 (2015) · Zbl 1331.93168 |
| [13] | Isbell, J. R., Birkhoff’s problem 111, Proceedings of the Americal Mathematical Society, 6, 2, 217-218 (1955) · Zbl 0066.09404 |
| [14] | Jönsson, U., Lecture notes on integral quadratic constraints (2001), Department of Mathematics, Royal Instutue of Technology (KTH): Department of Mathematics, Royal Instutue of Technology (KTH) Stockholm, Sweden |
| [15] | Jönsson, U., & Laiou, M. C. (1996). Stability analysis of systems with nonlinearities. In Proc. 35th IEEE conf. decision control, vol. 2 (pp. 2145-2150). |
| [16] | Kalman, R., Physical and mathematical mechanisms of instability in nonlinear automatic control systems, Transactions of ASME, 79, 3, 553-566 (1957) |
| [17] | Kharitenko, A.; Scherer, C., Time-varying Zames-Falb multipliers for LTI systems are superfluous, Automatica, Article 110577 pp. (2022) · Zbl 1502.93006 |
| [18] | Khong, S. Z., On integral quadratic constraints, IEEE Transactions on Automatic Control, 67, 3, 1603-1608 (2021) · Zbl 1537.93558 |
| [19] | Khong, S.; Kao, C.-Y., Converse theorems for integral quadratic constraints, IEEE Transactions on Automatic Control, 66, 8, 3695-3701 (2020) · Zbl 1471.93212 |
| [20] | Khong, S. Z.; Kao, C.-Y., Addendum to “Converse theorems for integral quadratic constraints”, IEEE Transactions on Automatic Control, 67, 1, 539-540 (2021) · Zbl 1480.93312 |
| [21] | Khong, S. Z.; van der Schaft, A., On the converse of the passivity and small-gain theorems for input-output maps, Automatica, 97, 58-63 (2018) · Zbl 1406.93297 |
| [22] | Khong, S. Z.; Su, L., On the necessity and sufficiency of the Zames-Falb multipliers for bounded operators, Automatica, 131, Article 109787 pp. (2021) · Zbl 1478.93491 |
| [23] | Khong, S.; Su, L.; Zhao, D., On the exponential convergence of input-output signals of nonlinear feedback systems (2022), arXiv preprint arXiv:2206.01945 |
| [24] | Lee, B.; Seiler, P., Finite step performance of first-order methods using interpolation conditions without function evaluations (2020), arXiv preprint arXiv:2005.01825 |
| [25] | Lessard, L.; Recht, B.; Packard, A., Analysis and design of optimization algorithms via integral quadratic constraints, SIAM Journal on Optimization, 26, 1, 57-95 (2016) · Zbl 1329.90103 |
| [26] | Megretski, A. (1995). Combining \(L_1\) and \(L_2\) methods in the robust stability and performance analysis of nonlinear systems. In Proceedings of 34th IEEE conference on decision and control, vol. 3 (pp. 3176-3181). [ISSN: 0191-2216]. |
| [27] | Megretski, A.; Rantzer, A., System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control, 42, 6, 819-830 (1997) · Zbl 0881.93062 |
| [28] | Megretski, A.; Treil, S., Power distribution inequalities in optimization and robustness of uncertain systems, Journal of Mathematical Systems, Estimation, and Control, 3, 3, 301-319 (1993) · Zbl 0781.93079 |
| [29] | Michalowsky, S.; Scherer, C.; Ebenbauer, C., Robust and structure exploiting optimisation algorithms: an integral quadratic constraint approach, International Journal of Control, 94, 11, 2956-2979 (2021) · Zbl 1481.90246 |
| [30] | O’Shea, R., An improved frequency time domain stability criterion for autonomous continuous systems, IEEE Transactions on Automatic Control, 12, 6, 725-731 (1967) |
| [31] | O’Shea, R.; Younis, M., A frequency-time domain stability criterion for sampled-data systems, IEEE Transactions on Automatic Control, 12, 6, 719-724 (1967) |
| [32] | Rantzer, A.; Megretski, A., System analysis via integral quadratic constraints: Part II (1997), Department of Automatic Control, Lund Institute of Technology (LTH) · Zbl 0881.93062 |
| [33] | Seiler, P.; Carrasco, J., Construction of periodic counterexamples to the discrete-time Kalman conjecture, IEEE Control Systems Letters, 5, 4, 1291-1296 (2020) |
| [34] | Turner, M. C.; Drummond, R., Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity, International Journal of Robust and Nonlinear Control, 31, 6, 2255-2273 (2021) · Zbl 1526.93140 |
| [35] | Wang, S.; Carrasco, J.; Heath, W. P., Phase limitations of Zames-Falb multipliers, IEEE Transactions on Automatic Control, 63, 4, 947-959 (2017) · Zbl 1390.93627 |
| [36] | Willems, J. C., (The analysis of feedback systems. The analysis of feedback systems, M.I.T. Press research monographs (1970), MIT Press) |
| [37] | Willems, J. C.; Brockett, R., Some new rearrangement inequalities having application in stability analysis, IEEE Transactions on Automatic Control, 13, 5, 539-549 (1968) |
| [38] | Young, N., An introduction to Hilbert space (1988), Cambridge University Press · Zbl 0645.46024 |
| [39] | Zames, G.; Falb, P. L., Stability conditions for systems with monotone and slope-restricted nonlinearities, SIAM Journal on Control, 6, 1, 89-108 (1968) · Zbl 0157.15801 |
| [40] | Zhang, J.; Carrasco, J.; Heath, W. P., Duality bounds for discrete-time Zames-Falb multipliers, IEEE Transactions on Automatic Control, 67, 7, 3521-3528 (2022) · Zbl 1537.93632 |
| [41] | Zhang, J.; Seiler, P.; Carrasco, J., Zames-Falb multipliers for convergence rate: motivating example and convex searches, International Journal of Control, 95, 3, 821-829 (2022) · Zbl 1485.93463 |
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