# 리아프노브 분석법 기반 비선형 적응제어 개요 및 연구동향 조사

• Park, Jin Bae (Department of Electrical and Electronic Engineering, Yonsei University) ;
• Lee, Jae Young (Department of Electrical and Electronic Engineering, Yonsei University)
• 박진배 (연세대학교 전기전자공학과) ;
• 이재영 (연세대학교 전기전자공학과)
• Accepted : 2014.02.03
• Published : 2014.03.01

#### Abstract

This paper provides an overview of the basics and recent studies of Lyapunov-based nonlinear adaptive control, the aim of which is to improve or maintain the performance and stability of the closed-loop system by cancelling out the presumable uncertainties in the nonlinear system dynamics. The design principles are essentially based on Lyapunov's direct method. In this survey, we provide a comprehensive overview of Lyapunov-based nonlinear adaptive control techniques with simplified effective design examples, which are to be elaborated as related recent results are gradually shown. The scope of the survey contains research on singularity problems in adaptive control, the techniques to deal with linearly and nonlinearly parameterized uncertainties, robust neuro-adaptive control, and adaptive control methodologies combined with various nonlinear control techniques such as sliding-mode control, back-stepping, dynamic surface control, and optimal/$H_{\infty}$ control.

#### References

1. J.-J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall New Jersey, 1991.
2. K. Zhou and J. C. Doyle, Essentials of Robust Control, Prentice Hall Upper Saddle River, NJ, 1998.
3. F. L. Lewis, Optimal Estimation: with an Introduction to Stochastic Control Theory, Wiley New York et al., 1986.
4. M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons NY, 1995.
5. K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, Inc., 1989.
6. P. A. Ioannou and J. Sun, Robust Adaptive Control, Prentice-Hall, Inc., 1989.
7. H. K. Khalil, Nonlinear Systems, Prentice hall Upper Saddle River, 2002.
8. X. Liu, R. Ortega, H. Su, and J. Chu, "Immersion and invariance adaptive control of nonlinearly parameterized nonlinear systems," IEEE Trans. Automat. Contr., vol. 55, no. 9, pp. 2209-2214, 2010. https://doi.org/10.1109/TAC.2010.2052389
9. A. Astolfi and R. Ortega, "Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems," IEEE Trans. Automat. Contr., vol. 48, no. 4, pp. 590-606, 2003. https://doi.org/10.1109/TAC.2003.809820
10. K. Hornik, M. Stinchcombe, and H. White, "Multilayer feedforward networks are universal approximators," Neural Netw., vol. 2, no. 5, pp. 359-366, 1989. https://doi.org/10.1016/0893-6080(89)90020-8
11. J. Park and I. W. Sandberg, "Universal approximation using radial-basis-function networks," Neural Comput., vol. 3, no.2, pp. 246-257, 1991. https://doi.org/10.1162/neco.1991.3.2.246
12. J. Zhang, G. G. Walter, Y. Miao, W. N. W. Lee, "Wavelet neural networks for function learning," IEEE Trans. Signal Process., vol. 43, no. 6, pp. 1485-1497, 1995. https://doi.org/10.1109/78.388860
13. J. T. Spooner and K. M. Passino, "Stable adaptive control using fuzzy systems and neural networks," IEEE Trans. Fuzzy Syst., vol. 4, no. 3, pp. 339-359, 1996. https://doi.org/10.1109/91.531775
14. T. Zhang, S. S. Ge, and C. C. Hang, "Stable adaptive control for a class of nonlinear systems using a modified Lyapunov function," IEEE Trans. Automat. Contr., vol. 45, no. 1, pp. 129-132, 2000. https://doi.org/10.1109/9.827369
15. A. Yesidirek and F. L. Lewis, "Feedback linearization using neural networks," Automatica, vol. 31, no. 11, pp. 1659-1664, 1995. https://doi.org/10.1016/0005-1098(95)00078-B
16. M. M. Polycarpou, "Stable adaptive neural control scheme for nonlinear systems," IEEE Trans. Automat. Contr., vol. 41, no. 3, pp. 447-451, 1996. https://doi.org/10.1109/9.486648
17. M. M. Polycarpou and M. J. Mears, "Stable adaptive tracking of uncertain systems using nonlinearly parameterized on-line approximators," Int. J. Control, vol. 70, no. 3, pp. 363-384, 1998. https://doi.org/10.1080/002071798222280
18. K. S. Narendra and A. M. Annaswamy, "A new adaptive law for robust adaptation without persistent excitation," IEEE Trans. Automat. Contr., vol. AC-32, pp. 134.145, Feb. 1987.
19. K. Y. Volyanskyy, W. M. Haddad, and A. J. Calise, "A new neuroadaptive control architecture for nonlinear uncertain dynamical systems: beyond ${\sigma}-$ and e-modifications," IEEE Trans. Neural Netw., vol. 20, no. 11, pp. 1707-1723, 2009. https://doi.org/10.1109/TNN.2009.2030748
20. S. M. Naik, P. R. Kumar, and B. E. Ydstie, "Robust continuous-time adaptive control by parameter projection," IEEE Trans. Automat. Contr., vol. 37, no. 2, pp. 182-197, 1992. https://doi.org/10.1109/9.121620
21. J.-J. E Slotine and J. A. Coetsee, "Adaptive sliding controller synthesis for non-linear systems," Int. J. Control, vol. 43, no. 6, pp. 1631-1635, 1986. https://doi.org/10.1080/00207178608933564
22. B. Yao and M. Tomizuka, "Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form," Automatica, vol. 33, no. 5, pp. 893-900, 1997. https://doi.org/10.1016/S0005-1098(96)00222-1
23. B. Yao, F. Bu, and G. T. C. Chiu, "Non-linear adaptive robust control of electro-hydraulic systems driven by double-rod actuators," Int. J. Control, vol. 74, no. 8, pp. 761-775, 2001. https://doi.org/10.1080/002071700110037515
24. H. Xu, M. D. Mirmirani, and P. A. Ioannou, "Adaptive sliding mode control design for a hypersonic flight vehicle," J. Guid., Control, Dyn., vol. 27, no. 5, pp. 829-838, 2004. https://doi.org/10.2514/1.12596
25. L. Fiorentini, A. Serrani, M. A. Bolender, and D. B. Doman, "Nonlinear robust adaptive control of flexible air-breathing hypersonic vehicles," J. Guid., Control, Dyn., vol. 32, no. 2, pp. 402-417, 2009. https://doi.org/10.2514/1.39210
26. D. Lee, H. J. Kim, and S. Sastry, "Feedback linearization vs. adaptive sliding mode control for a quadrotor helicopter," Int. J. Control, Autom., Syst. (IJCAS), vol. 7, no. 3, pp. 419-428, 2009. https://doi.org/10.1007/s12555-009-0311-8
27. B. S. Park, S. J. Yoo, J. B. Park, and Y. H. Choi, "Adaptive neural sliding mode control of nonholonomic wheeled mobile robots with model uncertainty," IEEE Trans. Control Syst. Technol., vol. 17, no. 1, pp. 207-214, 2009. https://doi.org/10.1109/TCST.2008.922584
28. M. Zeinali and L. Notash, "Adaptive sliding mode control with uncertainty estimator for robot manipulators," Mechanism and Machine Theory, vol. 45, no. 1, pp. 80-90, 2010. https://doi.org/10.1016/j.mechmachtheory.2009.08.003
29. S. Tong and H.-X. Li, "Fuzzy adaptive sliding-mode control for MIMO nonlinear systems," IEEE Trans. Fuzzy Syst., vol. 11, no. 3, pp. 354-360, 2003. https://doi.org/10.1109/TFUZZ.2003.812694
30. M. Zeinali and L. Notash, "Robust adaptive neural fuzzy controller with model uncertainty estimator for manipulators," Special Ed., CSME Trans., vol. 28, no. 2A, pp. 197-219, 2004.
31. B. Yoo, and W. Ham, "Adaptive fuzzy sliding mode control of nonlinear system," IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 315-321, 1998. https://doi.org/10.1109/91.669032
32. T. Zhang, S. S. Ge, and C. C. Hang, "Adaptive neural network control for strict-feedback nonlinear systems using backstepping design," Automatica, vol. 36, no. 12, pp. 1835-1846, 2000. https://doi.org/10.1016/S0005-1098(00)00116-3
33. A. J. Koshkouei and A. S. I. Zinober, "Adaptive backstepping control of nonlinear systems with unmatched uncertainty," Proc. of IEEE Conf. Decis. Control, pp. 4765-4770, 2000.
34. I. Kanellakopoulos, P. Kokotovic, and A. Morse, "Systematic design of adaptive controllers for feedback linearizable systems," IEEE Trans. Automat. Contr., vol. 36, no. 11, pp. 1241-1253, 1991. https://doi.org/10.1109/9.100933
35. H. Du, H. Shao, and P. Yao, "Adaptive neural network control for a class of low-triangularstructured nonlinear systems," IEEE Trans. Neural Netw., vol. 17, no. 2, pp. 509-514, 2006. https://doi.org/10.1109/TNN.2005.863403
36. C. Wang, D. J. Hill, S. S. Ge, and G. Chen, "An ISS-modular approach for adaptive neural control of pure-feedback systems," Automatica, vol. 42, no. 5, pp. 723-731, 2006. https://doi.org/10.1016/j.automatica.2006.01.004
37. D. Wang and J. Huang, "Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form," Automatica, vol. 38, no. 8, pp. 1365-1372, 2002. https://doi.org/10.1016/S0005-1098(02)00034-1
38. D. Swaroop, J. C. Cerdes, P. P. Yip, and J. K. Hedrick, "Dynamic surface control of nonlinear systems," Proc. of Am. Control Conf., pp. 3028-3036, 1997.
39. D. Wang and J. Huang, "Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form," IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195-202, 2005. https://doi.org/10.1109/TNN.2004.839354
40. S. J. Yoo, J. B. Park, and Y. H. Choi, "Adaptive dynamic surface control for stabilization of parametric strict-feedback nonlinear systems with unknown time delays," IEEE Trans. Automat. Contr., vol. 52, no. 12, pp. 2360-2365, 2007. https://doi.org/10.1109/TAC.2007.910715
41. S. J. Yoo, J. B. Park, and Y. H. Choi, "Adaptive dynamic surface control of flexible-joint robots using self-recurrent wavelet neural networks," IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 36, no. 6, pp. 1342-1355, 2006. https://doi.org/10.1109/TSMCB.2006.875869
42. J. L. Fausz, V.-S. Chellaboina, and W. Haddad, "Inverse optimal adaptive control for nonlinear uncertain systems with exogenous disturbances," Int. J. Adapt. Control Signal Process., vol. 14, no. 1, pp. 1-38, 2000. https://doi.org/10.1002/(SICI)1099-1115(200002)14:1<1::AID-ACS595>3.0.CO;2-B
43. W. Luo, Y. Chu, and K. Ling, "Inverse optimal adaptive control for attitude tracking of spacecraft," IEEE Trans. Automat. Contr., vol. 50, no. 11, pp. 1639-1654, Nov. 2005. https://doi.org/10.1109/TAC.2005.858694
44. Z. Li and M. Krstic, "Optimal design of adaptive tracking controllers for nonlinear systems," Automatica, vol. 33, no. 4, pp. 1459-1473, 1997. https://doi.org/10.1016/S0005-1098(97)00072-1
45. L. Sonneveldt, E. Van Oort, Q. P. Chu, and J. A. Mulder, "Comparison of inverse optimal and tuning functions designs for adaptive missile control," J. Guid., Control, Dyn., vol. 31, no. 4, pp. 1176-1182, 2008. https://doi.org/10.2514/1.31316
46. K. Dupree, P. M. Patre, M. Johnson, and W. E. Dixon, "Inverse optimal adaptive control of a nonlinear Euler-Lagrange system, part I: full state feedback," Proc. of IEEE Conf. Decis. Control Held Jointly with Chinese Control Conference, pp. 321-326, 2009.
47. K. Dupree, P. M. Patre, Z. D. Wilcox, and W. E. Dixon, "Asymptotic optimal control of uncertain nonlinear Euler-Lagrange system," Automatica, vol. 47, no. 1, pp. 99-107, 2011. https://doi.org/10.1016/j.automatica.2010.10.007
48. F.-Y. Wang, H. Zhang, and D. Liu, "Adaptive dynamic programming: an introduction," IEEE Comput. Intell. Mag., vol. 4, no. 2, pp. 39-47, 2009. https://doi.org/10.1109/MCI.2009.932261
49. F. L. Lewis and D. Liu, Reinforcement Learning and Approximate Dynamic Programming for Feedback Control, IEEE Press, John Wiley & Sons, 2013,
50. F. L. Lewis and D. Vrabie, "Reinforcement learning and adaptive dynamic programming for feedback control," IEEE Circuits Syst. Mag., vol. 9, no. 3, pp. 32-50, 2009. https://doi.org/10.1109/MCAS.2009.933854
51. T. Hanselmann, L. Noakes, and A. Zaknich, Continuoustime Adaptive Critics, IEEE Trans. Neural Netw., vol. 18, no. 3, pp. 631-647, 2007. https://doi.org/10.1109/TNN.2006.889499
52. S. Bhasin, R. Kamalapurkar, M. Johnson, K. G. Vamvoudakis, F. L. Lewis, and W. E. Dixon, "A novel actor-critic-identifier architecture for approximate optimal control of uncertain nonlinear systems," Automatica, vol. 49, no. 1, pp. 82-92, 2013. https://doi.org/10.1016/j.automatica.2012.09.019
53. K. G. Vamvoudakis, D. Vrabie, and F. L. Lewis, "Online adaptive algorithm for optimal control with integral reinforcement learning," Int. J. Robust Nonlinear Control, 2013.
54. J. Y. Lee, J. B. Park, and Y. H. Choi, "Integral Q-learning and explorized policy iteration for adaptive optimal control of continuous-time linear systems," Automatica, vol. 48, no. 11, pp. 2850-2859, 2012. https://doi.org/10.1016/j.automatica.2012.06.008
55. J. Y. Lee, J. B. Park, and Y. H. Choi, "Integral reinforcement learning with explorations for continuous-time nonlinear systems," Proc. of Int. Jt. Conf. Neural Netw. (IJCNN), pp. 1042-1047, 2012.
56. J.-H. Kim and F. L. Lewis, "Model-free $H_{\infty}$ control design for unknown linear discrete-time systems via Q-learning with LMI," Automatica, vol. 46, no. 8, pp. 1320-1326, 2010. https://doi.org/10.1016/j.automatica.2010.05.002
57. K. K. Tan, T. H. Lee, S. N. Huang, and F. M. Leu, "Adaptive predictive control of a class of SISO nonlinear systems," Dynamics and Control, vol. 11, no. 2, pp. 151-174, 2001. https://doi.org/10.1023/A:1012583811904

#### Cited by

1. Decentralized Robust Adaptive Control for Robot Manipulators with Input Torque Saturation vol.21, pp.12, 2015, https://doi.org/10.5302/J.ICROS.2015.15.0173
2. Chattering Free Sliding Mode Control of Upper-limb Rehabilitation Robot with Handling Subject and Model Uncertainties vol.21, pp.5, 2015, https://doi.org/10.5302/J.ICROS.2015.15.9020