Abstract
Issues concerning locality and robustness for persistently excited nonlinear systems can be addressed on the basis of a recently presented strict quadratic Lyapunov function. Such Lyapunov function is characterized by a “stable” linear time-invariant matrix differential equation, whose solution, at runtime, is available from direct integration of measured quantities. Several applications (even including experiments) are here presented to illustrate the benefits of such a new technical tool.
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Notes
- 1.
In the aforementioned operating conditions, according to [30] we have \(\phi _{ra} = M i_{sa}+c \phi _{rb}\), \(\phi _{rb} = M i_{sb}-c \phi _{ra}\), in terms of \(c = {T_{L}L_{r}}/({\phi _{ra}^{2}+\phi _{rb}^{2})}\).
- 2.
Take for instance the scalar signal
$$\begin{aligned} y(t)= & {} \theta _{1} \sin \left( \theta _{2}\cos (t)\right) \end{aligned}$$depending on two positive parameters \(\theta _{1}\) and \(\theta _{2}\).
References
Anderson, B.D.O.: Exponential stability of linear equations arising in adaptive identification. IEEE Trans. Autom. Control 22, 83–88 (1977)
Asheghan, M.M., Míguez, J.: Stability analysis and robust control of heart beat rate during treadmill exercise. Automatica 63, 311–320 (2016)
Barabanov, N., Ortega, R.: On global asymptotic stability of \(\dot{x}=-\phi (t)\phi ^{\rm T}(t)x\) with \(\phi \) not persistently exciting. Syst. Control Lett. 109, 24–29 (2017)
Besançon, G., Bornard, G., Hammouri, H.: Observer synthesis for a class of nonlinear control systems. Euro. J. Control 2, 176–192 (1996)
Burden, R.L., Faires, J.D.: Numerical Analysis. Thomson Brooks/Cole, Belmont, USA (2005)
Cavallo, A., De Maria, G., Natale, C., Pirozzi, S.: Robust control of flexible structures with stable bandpass controllers. Automatica 44, 1251–1260 (2008)
Consolini, L., Verrelli, C.M.: Learning control in spatial coordinates for the path-following of autonomous vehicles. Automatica 50, 1867–1874 (2014)
De Luca, A., Oriolo, G., Samson, C.: Feedback Control of a nonholonomic car-like robot. In: Laumond , J.-P. (ed.) Robot Motion Planning and Control, Lectures Notes in Control and Information Sciences, vol. 229, pp. 171–253. Springer, Berlin (1998)
Efimov, D., Fradkov, A.: Design of impulsive adaptive observers for improvement of persistency of excitation. Int. J. Adaptive Control Signal Process. 1–18 (2014)
Gu, W., Cai, S., Hu, Y., Zhang, H., Chen, H.: Trajectory planning and tracking control of a ground mobile robot: a reconstruction approach towards space vehicle. ISA Trans. (2018). https://doi.org/10.1016/j.isatra.2018.11.019
Loria, A., Panteley, E.: Uniform exponential stability of linear time-varying systems: revisited. Syst. Control Lett. 47, 13–24 (2002)
Maghenem, M.A., Loria, A.: Strict Lyapunov functions for time-varying systems with persistency of excitation. Automatica 78, 274–279 (2017)
Maghenem, M., Loria, A.: Lyapunov functions for persistently-excited cascaded time-varying systems: application to consensus. IEEE Trans. Autom. Control 62, 3416–3422 (2017)
Mazenc, F.: Strict Lyapunov functions for time-varying systems. Automatica 39, 349–353 (2003)
Marino, R., Tomei, P.: Global estimation of \(n\) unknown frequencies. IEEE Trans. Autom. Control 47, 1324–1328 (2002)
Marino, R., Tomei, P.: Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. IEEE Trans. Autom. Control 40, 1300–1304 (1995)
Marino, R., Santosuosso, G.L., Tomei, P.: Robust adaptive observers for nonlinear systems with bounded disturbances. IEEE Trans. Autom. Control 46, 967–972 (2001)
Marino, R., Tomei, P.: Output regulation of unknown stable linear systems. IEEE Trans. Autom. Control 60, 2213–2218 (2015)
Marino, R., Tomei, P.: Hybrid adaptive multi-sinusoidal disturbance cancellation. IEEE Trans. Autom. Control 62, 4023–4030 (2016)
Marino, R., Tomei, P., Verrelli, C.M.: A new flux observer for induction motors with on-line identification of load torque and resistances. In: 18th IFAC World Congress, vol. 18, pp. 6172–6177. Milano (2011)
Paradiso, M., Pietrosanti, S., Scalzi, S., Tomei, P., Verrelli, C.M.: Experimental heart rate regulation in cycle-ergometer exercises. IEEE Trans. Biomed. Eng. 60, 135–139 (2013)
Progromsky, A.Y., Matveev, A.S.: Stability analysis of PE systems via Steklov’s averaging technique. Int. J. Adaptive Control Signal Process. 31, 138–144 (2017)
Sastry, S., Bodson, M.: Adaptive Control—Stability, Convergence, and Robustness. Prentice Hall, Englewood Cliffs, New Jersey (1989)
Scalzi, S., Tomei, P., Verrelli, C.M.: Nonlinear control techniques for the hear rate regulation in treadmill exercises. IEEE Trans. Biomed. Eng. 59, 599–603 (2012)
Tomei, P., Verrelli, C.M.: A nonlinear adaptive speed tracking control for sensorless permanent magnet step motors with unknown load torque. Int. J. Adaptive Control Signal Process. 22, 266–288 (2008)
Tomei, P., Verrelli, C.M.: Observer-based speed tracking control for sensorless permanent magnet synchronous motors with unknown load torque. IEEE Trans. Autom. Control 56, 1484–1488 (2011)
Tomei, P., Verrelli, C.M.: Novel algorithms for the synchronization control of nonlinear systems. Int. J. Adaptive Control Signal Process. 30, 608–633 (2016)
Tomei, P., Verrelli, C.M.: Advances on adaptive learning control: the case of non-minimum phase linear systems. Syst. Control Lett. 115, 55–62 (2018)
Verrelli, C.M.: Fourier series expansion for synchronization of permanent magnet electric motors. Appl. Math. Comput. 217, 4502–4515 (2011)
Verrelli, C.M.: Insights on observability and identifiability properties of induction motors at steady-state. In: IFAC International Workshop on Adaptation and Learning in Control and Signal Processing ALCOSP, vol. 11, pp. 110–115. Caen, France (2013)
Verrelli, C.M., Savoia, A., Mengoni, M., Marino, R., Tomei, P., Zarri, L.: On-line identification of winding resistances and load torque in induction machines. IEEE Trans. Control Syst. Technol. 22, 1629–1637 (2014)
Verrelli, C.M., Tomei, P.: Non-anticipating Lyapunov functions for persistently excited nonlinear systems. IEEE Trans. Autom. Control 65, 2634–2639 (2020)
Verrelli, C.M., Tomei, P., Caminiti, G., Iellamo, F., Volterrani, M.: Nonlinear heart rate control in treadmill/cycle-ergometer exercises under the instability constraint. Automatica 127, (2021)
Verrelli, C.M., Tomei, P., Lorenzani, E., Fornari, R., Immovilli, F.: Further results on nonlinear tracking control and parameter estimation for induction motors. Control Eng. Practice 66, 116–125 (2017)
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Verrelli, C.M., Tomei, P., Pirozzi, S., Fabiani, S. (2022). Analysis of Persistently Excited Nonlinear Systems with Applications. In: Zattoni, E., Simani, S., Conte, G. (eds) 15th European Workshop on Advanced Control and Diagnosis (ACD 2019). ACD 2019 2018. Lecture Notes in Control and Information Sciences - Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-85318-1_8
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