Convergence Properties of Adaptive Systems and the Definition of Exponential Stability
Abstract
The convergence properties of adaptive systems in terms of excitation conditions on the regressor vector are well known. With persistent excitation of the regressor vector in model reference adaptive control the state error and the adaptation error are globally exponentially stable or, equivalently, exponentially stable in the large. When the excitation condition, however, is imposed on the reference input or the reference model state, it is often incorrectly concluded that the persistent excitation in those signals also implies exponential stability in the large. The definition of persistent excitation is revisited so as to address some possible confusion in the adaptive control literature. It is then shown that persistent excitation of the reference model only implies local persistent excitation (weak persistent excitation). Weak persistent excitation of the regressor is still sufficient for uniform asymptotic stability in the large, but not exponential stability in the large. We show that there exists an infinite region in the state-space of adaptive systems where the state rate is bounded. This infinite region with finite rate of convergence is shown to exist not only in classic open-loop reference model adaptive systems but also in a new class of closed-loop reference model adaptive systems.
1. . Anderson, Exponential stability of linear equations arising in adaptive identification , IEEE Trans. Automat. Control , 22 ( 1977 ), pp. 83 -- 88 .
2. , Jr., Exponential convergence of adaptive identification and control algorithms , Automatica , 18 ( 1982 ), pp. 1 -- 13 .
3. , New results in linear system stability , SIAM J. Control , 7 ( 1969 ), pp. 398 -- 414 .
4. , On the stability of motion as a whole , Dokl. Akad. Nauk SSSR , 86 ( 1952 ), pp. 453 -- 456 .
5. , On parameter convergence in adaptive control , Systems Control Lett. , 3 ( 1983 ), pp. 311 -- 319 .
6. , Necessary and sufficient conditions for parameter convergence in adaptive control , Automatica , 22 ( 1986 ), pp. 629 -- 639 .
7. , Linearity vs. nonlinearity and asymptotic stability in the large , IEEE Trans. Circuit Theory , 12 ( 1965 ), pp. 117 -- 118 , https://doi.org/10.1109/TCT.1965.1082383.
8. , Adaptive output feedback based on closed-loop reference models , in IEEE Trans. Automat. Control , 60 ( 2015 ), pp. 2728 -- 2733 , https://doi.org/10.1109/TAC.2015.2405295.
9. T. E. Gibson, Closed-Loop Reference Model Adaptive Control: With Application to Very Flexible Aircraft, Ph.D. thesis, Massachusetts Institute of Technology, 2014.
10. T. E. Gibson and A. M. Annaswamy, Adaptive control and the definition of exponential stability, in Proceedings of the 2015 American Control Conference, 2015, pp. 1549--1554, https://doi.org/10.1109/ACC.2015.7170953.
11. T. E. Gibson, A. M. Annaswamy, and E. Lavretsky, Improved transient response in adaptive control using projection algorithms and closed loop reference models, in Proceedings of the AIAA Guidance Navigation and Control Conference, 2012.
12. T. E. Gibson, A. M. Annaswamy, and E. Lavretsky, Closed-loop reference model adaptive control, Part I: Transient performance, in Proceedings of the American Control Conference, 2013.
13. T. E. Gibson, A. M. Annaswamy, and E. Lavretsky, Closed-loop reference models for output--feedback adaptive systems, in Proceedings of the European Control Conference, 2013.
14. , On adaptive control with closed-loop reference models: Transients, oscillations, and peaking , IEEE Access , 1 ( 2013 ), pp. 703 -- 717 .
15. I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning, MIT Press, Cambridge, MA, 2016, http://www.deeplearningbook.org.
16. , Generative adversarial nets , in Advances in Neural Information Processing Systems , 2014 , pp. 2672 -- 2680 .
17. W. Hahn, Stability of Motion, Springer-Verlag, New York, 1967.
18. P. Ioannou and J. Sun, Robust Adaptive Control, Dover, New York, 2013.
19. B. Jenkins, T. Gibson, A. Annaswamy, and E. Lavretsky, Convergence properties of adaptive systems with open-and closed-loop reference models, in Proceedings of the AIAA Guidance Navigation and Control Conference, 2013.
20. , Uniform asymptotic stability and slow convergence in adaptive systems , in Proceedings of the IFAC International Workshop on Adaptation and Learning in Control and Signal Processing , Vol. 11 , 2013 , pp. 446 -- 451 .
21. , Contributions to the theory of optimal control , Bol. Soc. Math. Mex. , 5 ( 1960 ), pp. 102 -- 119 .
22. , Control systems analysis and design via the `second method' of Liapunov, I. Continuous-time systems , J. Basic Engineering , 82 ( 1960 ), pp. 371 -- 393 .
23. N. Krasovskii, Stability of Motion, Stanford University Press, Stanford, CA, 1963.
24. , Adaptive observers with exponential rate of convergence , IEEE Trans. Automat. Control , 22 ( 1977 ), pp. 2 -- 8 .
25. , Some extensions of Liapunov's second method , IRE Trans. Circuit Theory , 7 ( 1960 ), pp. 520 -- 527 , https://doi.org/10.1109/TCT.1960.1086720.
26. , Asymptotic stability criteria, in Hydrodynamic Instability, Proc. Sympos. Appl. Math. 13, AMS, Providence , RI , 1962 , pp. 299 -- 307 .
27. , Rapid identification of linear and nonlinear systems , AIAA J. , 5 ( 1967 ), pp. 1835 -- 1842 .
28. , Uniform exponential stability of linear time-varying systems: Revisited , Systems Control Lett. , 47 ( 2002 ), pp. 13 -- 24 .
29. , On stability in the first approximation , Sb. Nauch. Trudov Kazanskogo Aviac. Inst. , 3 ( 1935 ).
30. , Contributions to stability theory , Ann. Math. , 64 ( 1956 ), pp. 182 -- 206 .
31. , On the stability of nonautonomous differential equations ${\dot x = [{A}+{B}(t)]x}$, with skew symmetric matrix ${B(t)}$ , SIAM J. Control Optim. , 15 ( 1977 ), pp. 163 -- 176 .
32. , On the uniform asymptotic stability of certain linear nonautonomous differential equations , SIAM J. Control Optim. , 15 ( 1977 ), pp. 5 -- 24 .
33. , Robust adaptive control in the presence of bounded disturbances , IEEE Trans. Automat. Control , 31 ( 1986 ), pp. 306 -- 315 .
34. , Persistent excitation in adaptive systems , Internat. J. Control , 45 ( 1987 ), pp. 127 -- 160 .
35. K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, Dover, New York, 2005.
36. , Analysis of slow convergence regions in adaptive systems, in 2016 American Control Conference (ACC), Boston , MA , 2016 , pp. 6995 -- 7000 .
37. , Relaxed persistency of excitation for uniform asymptotic stability , IEEE Trans. Automat. Control , 46 ( 2001 ), pp. 1874 -- 1886 .
38. . de Freitas, Taking the human out of the loop: A review of bayesian optimization , Proc. IEEE , 104 ( 2016 ), pp. 148 -- 175 .
39. , Reinforcement learning is direct adaptive optimal control , IEEE Control Systems , 12 ( 1992 ), pp. 19 -- 22 .
40. , Probing signals for model reference identification , IEEE Trans. Automat. Control , 22 ( 1977 ), pp. 530 -- 538 .