Intelligent control of mechatronic systems

Adaptive Neural Network Control and its Applications Pey Yuen Tao B. Eng (Hons.), The National University of Singapore A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NUS GRADUATE SCHOOL FOR INTEGRATIVE SCIENCES & ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 Acknowledgments First of all, to my thesis supervisor, Professor Shuzhi Sam Ge, for all the learning opportunities, his unwavering support, inspiring guidance, and for his time and effort that he has spent on my training and education. I would like to thank Dr Wei Lin and Dr Xiaoqi Chen, my current and former thesis co-supervisors, for their guidance and help on all matters concerning my research. I would also like to thank Professor Jianxin Xu, for his insightful advice and guidance. To fellow co-workers and friends in the Mechatronics and Automation Lab, and the Edutainment Robotics Lab. Special thanks to Dr Cheng Heng Fua and Mr Keng Peng Tee, for the interesting discussions and time we spent working together. To the TechX team, in particular, Mr Aswin Thomas Abraham, Mr Brice Rebsamen, Mr Chenguang Yang, Ms Bahareh Ghotbi, Mr Dong Huang, Mr Hooman Samani, Dr Qinghua Xia and many others that have been part of the team, for the stressful but exciting time we have spent developing and testing the system. To Professor Hongbin Du from East China University of Science and Technology, Professor Tianping Zhang from Yangzhou University and Professor Zhijun Li from Shanghai Jiao Tong University, for the many enlightening discussions and help they have provided in my research. To Dr Feng Guan, Dr Xuecheng Lai, Dr Zhuping Wang, Mr Voon Ee How, Mr Sie Chyuan Lau, Ms Beibei Ren, Ms Yaozhang Pan, and many more, for their friendship and help. To my family and close friends for their support and encouragement. They have always been there for me, stood by me through the good times and the bad. Finally, I am very grateful to the Agency of Science, Technology and Research (A*STAR) and the NUS Graduate School of Integrative Sciences and Engineering (NGS), for their funding and support. ii Abstract This thesis considers the theoretical aspects of adaptive neural network (NN) control and its applications. Traditional adaptive controls are limited to systems where the dynamics can be expressed in the linear-in-parameters form, while NNs can approximate, to an arbitrary degree of accuracy, any real continuous function on a compact set. Through exploiting the approximation capabilities of the NNs, a NN based control can be developed where NNs are used to compensate for the functional and parametric uncertainties in the system model. In this thesis, the focus is on the control problem of a class of uncertain nonlinear pure-feedback single-input single-output (SISO) systems. The non-affine nonlinear control problem is challenging due to the relatively fewer mathematical tools available in comparison with that for affine nonlinear systems. In particular, it is difficult to construct control laws because pure-feedback systems have no affine appearance of the variables which can be used as virtual controls. In this thesis, an Adaptive Variable Structure Control (AVSC) is proposed, based on NN parametrization, for the class of uncertain nonlinear pure-feedback single-input single-output (SISO) systems. For the design of the AVSC, three backstepping approaches are explored in this thesis where the comparative advantages are highlighted during the design stages and based on the simulation studies. First, decoupled backstepping is proposed due to the possibility of a circular control design if traditional backstepping methods are used to cancel the coupled terms since the virtual control coefficients are still non-affine. Subsequently, a coupled backstepping approach is adopted where the coupled terms are handled in the next step in order to remove an assumption made on the system that was required in decoupled backstepping. Finally, the design of the AVSC with dynamic iii surface control (DSC) is explored to further reduce the size of the NNs required. In the subsequent chapter, the application of intelligent control is studied, where the focus is on the control of marine shafting systems. A case study on the application of intelligent control in marine shafting systems modeled as a chained multiple mass-spring system is presented where the control objectives include the tracking of the desired trajectory and the reduction of torsional vibrations within the shafting system. In the application study, the control design is closely coupled with the system model. Therefore, the system model for the marine shafting system is first presented where the system model contains functional and parametric uncertainties. The functional uncertainties include the hydrodynamic forces acting on the propeller, which are highly nonlinear and subjected to variations due to the diverse operating conditions such as air suction, cavitation and partial/full emergence of the propellers, and the frictional forces acting on the shafting system. The parametric uncertainties include the unknown moment of inertia of each mass unit and torsional stiffness of the massless springs connecting the mass units. Considering the dynamic model of the system and the uncertainties, an adaptive NN control is proposed where NNs are used to approximate the functional and parametric uncertainties in the system. In this application study, two adaptive NN controls are developed for the marine shafting system, where the first control is uses the decoupled backstepping approach followed by a second control design where the DSC approach is adopted. A simulation study is conducted to illustrate the effects of the two proposed controls and to analyze the difference in the two approaches in particular on the implementation and performance issues. iv Contents Acknowledgments ii Abstract iii Table of Contents vi List of Figures vi Introduction 1.1 Adaptive Variable Structure Control . . . . . . . . . . . . . . . . . . . 1.2 Adaptive NN Control of Marine Shafting System . . . . . . . . . . . . 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AVSC with Decoupled Backstepping 2.1 Introduction . . . . . . . . . . . . . . . 2.2 Problem Formulation and Preliminaries 2.3 Neural Networks and Parametrization . 2.4 Control Design . . . . . . . . . . . . . 2.5 Stability Analysis . . . . . . . . . . . . 2.6 Simulation . . . . . . . . . . . . . . . . AVSC with Coupled Backstepping 3.1 Introduction . . . . . . . . . . . . . . . 3.2 Problem Formulation and Preliminaries 3.3 Control Design . . . . . . . . . . . . . 3.4 Stability Analysis . . . . . . . . . . . . 3.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10 11 15 17 27 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 46 47 53 55 AVSC with Dynamic Surface Control 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 Problem Formulation and Preliminaries . . . . 4.3 Neural Networks for Dynamic Surface Control 4.4 Control Design . . . . . . . . . . . . . . . . . 4.5 Stability Analysis . . . . . . . . . . . . . . . . 4.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 71 72 73 74 87 91 v . . . . . . . . . . . . . . . Contents Adaptive NN Control of Marine Shafting System 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dynamic Modeling . . . . . . . . . . . . . . . . . . . . . . 5.3 Adaptive NN Control with Decoupled Backstepping . . . . 5.4 Adaptive NN Control with DSC . . . . . . . . . . . . . . . 5.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Adaptive NN Control with Decoupled Backstepping 5.5.2 Adaptive NN Control with DSC . . . . . . . . . . . Conclusion and Future Work 6.1 Summary and Contributions . . . . . . . . . . . . . . . 6.1.1 Adaptive Variable Structure Control . . . . . . . 6.1.2 Adaptive NN Control of Marine Shafting System 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 106 107 108 119 134 135 139 . . . . 145 145 146 147 148 Bibliography 149 A Derivations for Simulation 157 B Author’s Publications 165 vi List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 Neural network compact sets . . . . . . . . . . . . . . . . . . . . . . . State variables x1 , x2 , and x3 under AVSC with decoupled backstepping Control effort u under AVSC with decoupled backstepping . . . . . . . Adaptive parameters ψ1 , ψ2 , and ψ3 under AVSC with decoupled backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System output y = x1 and desired trajectory yd under AVSC with decoupled backstepping for CSTR . . . . . . . . . . . . . . . . . . . . . System state x2 under AVSC with decoupled backstepping for CSTR . Tracking error y − yd under AVSC with decoupled backstepping for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control effort u under AVSC with decoupled backstepping for CSTR . Adaptive parameters ψ1 and ψ2 under AVSC with decoupled backstepping for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 38 38 State variables x1 , x2 , and x3 under AVSC with coupled backstepping . Control effort u under AVSC with coupled backstepping . . . . . . . . Adaptive parameters ψ1 , ψ2 , and ψ3 under AVSC with coupled backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System output y = x1 and desired trajectory yd under AVSC with coupled backstepping for CSTR . . . . . . . . . . . . . . . . . . . . . . . System state x2 under AVSC with coupled backstepping for CSTR . . . Tracking error y − yd under AVSC with coupled backstepping for CSTR Control effort u under AVSC with coupled backstepping for CSTR . . Parameter ψ1 under AVSC with coupled backstepping for CSTR . . . . Parameter ψ2 under AVSC with coupled backstepping for CSTR . . . . System output y = x1 and desired trajectory yd under AVSC with coupled backstepping and large control gains for CSTR . . . . . . . . . . System state x2 under AVSC with coupled backstepping and large control gains for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracking error y − yd under AVSC with coupled backstepping and large control gains for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . Control effort u under AVSC with coupled backstepping and large control gains for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive parameter ψ1 under AVSC with coupled backstepping and large control gains for CSTR . . . . . . . . . . . . . . . . . . . . . . . 58 58 vii 39 42 43 43 44 44 59 61 61 62 62 63 63 64 64 65 65 66 List of Figures 3.15 Adaptive parameter ψ2 under AVSC with coupled backstepping and large control gains for CSTR . . . . . . . . . . . . . . . . . . . . . . 3.16 System output y = x1 and desired trajectory yd under AVSC with coupled backstepping and integral action for CSTR . . . . . . . . . . . . 3.17 System state x2 under AVSC with coupled backstepping and integral action for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Tracking error y − yd under AVSC with coupled backstepping and integral action for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Control effort u under AVSC with coupled backstepping and integral action for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Adaptive parameters ψ1 and ψ2 under AVSC with coupled backstepping and integral action for CSTR . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 5.1 5.2 5.3 5.4 5.5 . 66 . 68 . 69 . 69 . 70 . 70 State variables x1 , x2 , and x3 under AVSC with dynamic surface control Control effort u under AVSC with dynamic surface control . . . . . . . Adaptive parameters ψ1 , ψ2 , and ψ3 under AVSC with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error signal y1 = ω1 − α1 under AVSC with dynamic surface control . Error signal y2 = ω2 − α2 under AVSC with dynamic surface control . System output y = x1 and desired trajectory yd under AVSC with DSC for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System state x2 under AVSC with DSC for CSTR . . . . . . . . . . . . Tracking error y − yd under AVSC with DSC for CSTR . . . . . . . . Control effort u under AVSC with DSC for CSTR . . . . . . . . . . . Adaptive parameter ψ1 under AVSC with DSC for CSTR . . . . . . . . Adaptive parameter ψ2 under AVSC with DSC for CSTR . . . . . . . . System output y = x1 and desired trajectory yd under AVSC with DSC and integral action for CSTR . . . . . . . . . . . . . . . . . . . . . . . System state x2 under AVSC with DSC and integral action for CSTR . Tracking error y − yd under AVSC with DSC and integral action for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control effort u under AVSC with DSC and integral action for CSTR . Adaptive parameters ψ1 and ψ2 under AVSC with DSC and integral action for CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propeller velocity tracking performance under Adaptive NN control with decoupled backstepping . . . . . . . . . . . . . . . . . . . . . . Tracking Error under Adaptive NN control with decoupled backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsional vibrations in the first seconds under Adaptive NN control with decoupled backstepping . . . . . . . . . . . . . . . . . . . . . . Control effort u under Adaptive NN control with decoupled backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsional vibrations in shafting system with constant control effort u = 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 93 94 94 95 95 97 97 98 98 99 99 103 103 104 104 105 . 137 . 137 . 138 . 138 . 139 List of Figures 5.6 Propeller velocity tracking performance under Adaptive NN control with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . 5.7 Tracking Error under Adaptive NN control with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Torsional vibrations in the first seconds under Adaptive NN control with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . 5.9 Control effort u under Adaptive NN control with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Error signal y1 = ω1 − α1 under Adaptive NN control with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Error signal y2 = ω2 − α2 under Adaptive NN control with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Error signal y3 = ω3 − α3 under Adaptive NN control with dynamic surface control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix . 141 . 141 . 142 . 142 . 143 . 143 . 144 Chapter Introduction Demand for improved control performance and the prevalence of increasingly complex systems have led to extensive research in the field of control theory to develop novel control methodologies or to extend existing control designs for a larger class of systems and to compensate for the uncertainties in the system model. The control design is critical to the performance of the closed-loop system where poor design can lead to poor performance or instability. Therefore, rigorous stability analysis of the closedloop system is essential to ensure that all the signals in the system remain bounded. The development of a suitable system model is also essential in practical applications and the performance of the closed-loop system is highly dependent on the system model and control approach adopted. Ignoring modeling uncertainties during control design can result in poor performance and may even lead to instability of the closedloop system. While many control methods have been developed, practical application of these controls have to be adapted based on the target system and the control design has to be closely coupled with the system model in order to achieve the desired performance. In addition, by drawing on engineering insights and physical properties of the system, it is possible to develop a more elegant control design for very complex systems. In this chapter, an overview of the motivation and background of the research conducted on intelligent control is first presented, followed by an outline of the thesis. Bibliography [13] H. K. Khalil, “Adaptive output feedback control of nonlinear ststems represented by input-output models,” IEEE Transcations on Automatic Control, vol. 41, no. 2, pp. 177–188, 1996. [14] P. A. Ioannou and P. V. Kokotovi´c, Adaptive Systems with Reduced Models. New York: Springer-Verlag, 1983. [15] K. Narendra and A. Annaswamy, “A new adaptive law for robust adaptation without persistent excitation,” IEEE Transactions on Automatic Control, vol. 32, no. 2, pp. 134–145, 1987. [16] ——, Stable Adaptive Systems. Eaglewood Cliffs, NJ: Prentice-Hall, 1989. [17] A. Isidori, Nonlinear Control Systems, 3rd ed. London: Springer-Verlag, 1995. [18] G. Campion and G. Bastin, “Indirect adaptive state feedback control of linearly parametrized non-linear systems,” International Journal of Adaptive Control and Signal Processing, vol. 4, no. 5, pp. 345–358, 1990. [19] S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Transactions on Automatic Control, vol. 34, no. 11, pp. 1123–1131, 1989. [20] D. G. Taylor, P. V. Kokotovi´c, R. Marino, and I. Kannellakopoulos, “Adaptive regulation of nonlinear systems with unmodeled dynamics,” IEEE Transactions on Automatic Control, vol. 34, no. 4, pp. 405–412, 1989. [21] J.-B. Pomet and L. Praly, “Adaptive nonlinear regulation: estimation from the lyapunov equation,” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 729–740, 1992. [22] T. Johansen and P. A. Ioannou, “Robust adaptive control of minimum phase nonlinear systems,” International Journal of Adaptive Control and Signal Processing, vol. 10, no. 1, pp. 61–78, 1996. [23] I. Kanellakopoulos, P. V. Kokotovi´c, and R. Marino, “An extended direct scheme for robust adaptive nonlinear control,” Automatica, vol. 27, no. 2, pp. 247–255, 1991. [24] I. Kanellakopoulos, P. V. Kokotovi´c, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Transactions on Automatic Control, vol. 36, no. 11, pp. 1241–1253, 1991. [25] P. V. Kokotovi´c, Foundations of Adaptive Control. New York: Springer-Verlag, 1991. [26] M. Krsti´c, I. Kanellakopoulos, and P. V. Kokotovi´c, “Adaptive nonlinear control without overparametrization,” Systems & Control Letters, vol. 19, no. 3, pp. 177– 185, 1992. 150 Bibliography [27] ——, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [28] P. V. Kokotovi´c and M. Arcak, “Constructive nonlinear control: a historical perspective,” Automatica, vol. 37, no. 5, pp. 637–662, 2001. [29] D. Seto, A. M. Annaswamy, and J. Baillieul, “Adaptive control of nonlinear systems with a triangular structure,” IEEE Transactions on Automatic Control, vol. 39, no. 7, pp. 1411–1428, 1994. [30] M. M. Polycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 32, no. 3, pp. 423–427, 1996. [31] Z. Pan and T. Basar, “Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems,” IEEE Transactions on Automatic Control, vol. 43, no. 8, pp. 1066–1083, 1998. [32] M. Krsti´c and P. V. Kokotovi´c, “Adaptive nonlinear output-feedback schemes with marino-tomei controller,” IEEE Transactions on Automatic Control, vol. 41, no. 2, pp. 274–280, 1996. [33] M. Jankovic, “Adaptive nonlinear output feedback tracking with a partial highgain observer and backstepping,” IEEE Transactions on Automatic Control, vol. 42, no. 1, pp. 106–113, 1997. [34] H. K. Khalil, Nonlinear Systems. New Jersey: Prentice-Hall, 2002. [35] J. D. Boskovic, “Stable adaptive control of a class of first-order nonlinearly parameterized plants,” IEEE Transactions on Automatic Control, vol. 40, no. 2, pp. 347–350, 1995. [36] ——, “Adaptive control of a class of nonlinearly parameterized plants,” IEEE Transactions on Automatic Control, vol. 43, no. 7, pp. 930–934, 1998. [37] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Transactions on Neural Networks, vol. 1, no. 1, pp. 4–7, 1990. [38] K. Narendra and K. Parthasarathy, “Gradient methods for the optimization of dynamical systems containing neural networks,” IEEE Transactions on Neural Networks, vol. 2, no. 2, pp. 252–262, 1991. [39] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Transactions on Neural Networks, vol. 3, no. 6, pp. 837–863, 1992. [40] X. Cui and K. G. Shin, “Direct control and coordination using neural networks,” IEEE Transactions on Systems, Man and Cybernetics, vol. 23, no. 4, pp. 686– 697, 1993. 151 Bibliography [41] F. L. Lewis, K. Liu, and A. Yes¸ildirek, “Neural net robot controller with guaranteed tracking performance,” IEEE Transactions on Neural Networks, vol. 6, no. 2, pp. 703–715, 1995. [42] F. L. Lewis, A. Yes¸ildirek, and K. Liu, “Multilayer neural-net robot controller with guaranteed tracking performance,” IEEE Transactions on Neural Networks, vol. 7, no. 2, pp. 388–399, 1996. [43] A. Yesildirek and F. L. Lewis, “Feedback linearization using neural networks,” Automatica, vol. 31, no. 11, pp. 1659–1664, 1995. [44] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 447–451, 1996. [45] M. M. Polycarpou and M. J. Mears, “Stable adaptive tracking of uncertain systems using nonlinearly parametrized on-line approximators,” International Journal of Control, vol. 70, no. 3, pp. 363–384, 1998. [46] T. Zhang, S. S. Ge, and C. C. Hang, “Design and performance analysis of a direct adaptive controller for feedback linearizable systems,” in Proceedings of the American Control Conference, vol. 6, Philadelphia, Pennsylvania, June 1998, pp. 3746–3750. [47] S. S. Ge, C. C. Hang, and T. Zhang, “A direct method for robust adaptive nonlinear control with guaranteed transient performance,” Systems & Control Letters, vol. 37, no. 5, pp. 275–284, 1999. [48] ——, “A direct adaptive controller for dynamic systems with a class of nonlinear parameterizations,” Automatica, vol. 35, no. 4, pp. 741–747, 1999. [49] C. Kwan and F. L. Lewis, “Robust backstepping control of nonlinear systems using neural networks,” IEEE Transactions on Systems, Man and Cybernetics Part A: Systems and Human, vol. 30, no. 6, pp. 753–766, 2000. [50] G. A. Rovithakis, “Stable adaptive neuro-control design via lyapunov function derivative estimation,” Automatica, vol. 37, no. 8, pp. 1213–1221, 2001. [51] K. Hornik, “Approximation capabilities of multilayer feedforward networks,” Neural Networks, vol. 4, no. 2, pp. 251–257, 1991. [52] D. Gorinevsky, “On the persistency of excitation in radial basis function network identification of nonlinear systems,” IEEE Transactions on Neural Networks, vol. 6, no. 5, pp. 1237–1244, 1995. [53] E. B. Kosmatopoulos, M. M. Polycarpou, M. A. Christodoulou, and P. A. Ioannou, “High-order neural network structures for identification of dynamical systems,” IEEE Transactions on Neural Networks, vol. 6, no. 2, pp. 422–431, 1995. 152 Bibliography [54] Y. Zhang, P.-Y. Peng, and Z.-P. Jiang, “Stable neural controller design for unknown nonlinear systems using backstepping,” IEEE Transactions on Neural Networks, vol. 11, no. 6, pp. 1347–1360, 2000. [55] G. Arslan and T. Bas¸ar, “Disturbance attenuating controller design for strictfeedback systems with structurally unknown dynamics,” Automatica, vol. 37, no. 8, pp. 1175–1188, 2001. [56] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Norwell, USA: Kluwer Academic, 2001. [57] J. Q. Gong and B. Yao, “Neural network adaptive robust control of nonlinear systems in semi-strict feedback form,” Automatica, vol. 37, no. 8, pp. 1149– 1160, 2001. [58] N. Hovakimyan, F. Nardi, and A. J. Calise, “A novel error observer-based adaptive output feedback approach for control of uncertain systems,” IEEE Transactions on Automatic Control, vol. 47, no. 8, pp. 1310–1314, 2002. [59] S. S. Ge and C. Wang, “Adaptive nn control of uncertain nonlinear pure-feedback systems,” Automatica, vol. 38, pp. 671–682, 2002. [60] 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. [61] S. S. Ge, C. C. Hang, and T. Zhang, “Adaptive neural network control of nonlinear systems by state and output feedback,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, vol. 29, no. 6, pp. 818–828, 1999. [62] S. S. Ge and J. Zhang, “Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback,” IEEE Transactions on Neural Networks, vol. 14, no. 4, pp. 900–918, 2003. [63] C. Wang, D. J. Hill, and S. S. Ge, “Adaptive neural control of non-affine purefeedback systems,” in Proceedings of the IEEE International Symposium on Intelligent Control, Limassol, Cyprus, June 2005, pp. 298–303. [64] H. Du, H. Shao, and P. Yao, “Adaptive neural network control for a class of low-triangular-structured nonlinear systems,” IEEE Transactions on Neural Networks, vol. 17, no. 2, pp. 509–514, 2006. [65] D. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes, “Dynamic surface control for a class of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 45, no. 10, pp. 1893–1899, 2000. [66] D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form,” IEEE Transactions on Neural Networks, vol. 16, no. 1, pp. 195–202, 2005. 153 Bibliography [67] B. Song and J. K. Hedrick, “Observer-based dynamic surface control for a class of nonlinear systems: An lmi approach,” IEEE Transactions on Automatic Control, vol. 49, no. 11, pp. 1995–2001, 2004. [68] J. Green and J. K. Hedrick, “Nonlinear speed control for automotive engines,” in Proceedings of the 1990 American Control Conference, vol. 3, San Diego, CA, 1990, pp. 2891–2898. [69] S. J. Yoo, J. B. Park, , and Y. H. Choi, “Adaptive neural dynamic surface control of nonlinear time-delay systems with model uncertainties,” in Proceedings of the 2006 American Control Conference, Minneapolis, Minnesota, Jun 2006, pp. 3140–3145. [70] ——, “Adaptive output feedback control of flexible-joint robots using neural networks: dynamic surface design approach,” IEEE Transactions on Neural Networks, vol. 19, no. 10, pp. 1712–1726, 2008. [71] R. Marino and P. Tomei, “Global adaptive output-feedback control of nonlinear systems. I. linear parameterization,” IEEE Transactions on Automatic Control, vol. 38, no. 1, pp. 17–32, 1993. [72] ——, “Global adaptive output-feedback control of nonlinear systems. II. nonlinear parameterization,” IEEE Transactions on Automatic Control, vol. 38, no. 1, pp. 33–48, 1993. [73] P. A. Ioannou and B. Fidan, Adaptive Control Tutorial. 2006. Philadelphia: SIAM, [74] K. Nam and A. Araposthathis, “A model reference adaptive control scheme for pure-feedback nonlinear systems,” IEEE Transactions on Automatic Control, vol. 33, no. 9, pp. 803–811, 1988. [75] R. Marino and P. Tomei, “Global adaptive observers for nonlinear systems via filtered transformations,” IEEE Transactions on Automatic Control, vol. 37, no. 8, pp. 1239–1245, 1992. [76] W. Dayawansa, W. M. Boothby, and D. L. Elliott, “Global state and feedback equivalence of nonlinear systems,” Systems & Control Letters, vol. 6, no. 4, pp. 229–234, 1985. [77] L. Hunt, R. Su, and G. Meyer, “Global transformations of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 28, no. 1, pp. 24–31, 1983. [78] R. Marino and P. Tomei, “Dynamic output feedback linearization and global stabilization,” Systems & Control Letters, vol. 17, no. 2, pp. 115–121, 1991. ˚ om, “Theory and applications of adaptive control - a survey,” Automat[79] K. J. Astr¨ ica, vol. 19, no. 5, pp. 471–486, 1983. 154 Bibliography [80] W. K. Wilson, Practical solution of torsional vibration problems. Chapman & Hall, 1971. London: [81] J. S. Wu and W. H. Chen, “Computer method for forced torsional vibration of propulsive shafting system of marine engine with or without damping,” Journal of Ship Research, vol. 26, no. 3, pp. 176–189, 1982. [82] K. Bruun, E. Pedersen, and H. Valland, “The development of vehicular powertrain system modeling methodologies: Philosophy and implementation,” in Proceedings of the International Conference on Bond Graph Modeling, 2005. [83] T. I. Fossen and M. Blanke, “Nonlinear output feedback control of underwater vehicle propellers using feedback form estimated axial flow velocity,” IEEE Journal of Oceanic Engineering, vol. 25, no. 2, pp. 241–255, 2000. ˚ [84] A. J. Sørensen, A. K. Adnanes, T. I. Fossen, and J. P. Strand, “A new method of thruster control in positioning of ships based on power control,” in Proceedings of the IFAC Conference on Manoeuvering and Control of Marine Craft, Brijuni, Croatia, Sept 1997, pp. 172–179. [85] J. P. Breslin and P. Andersen, Hydrodynamics of Ship Propellers. University Press, 1994. Cambridge: [86] S. S. Ge, J. Zhang, and T. Lee, “Adaptive neural network control for a class of mimo nonlinear systems with disturbances in discrete-time,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, vol. 34, no. 4, pp. 1630– 1645, 2004. [87] T. M. Apostol, Mathematical Analysis. Reading, MA: Addison-Wesley, 1974. [88] S. S. Ge, “Adaptive control of uncertain lorenz system using decoupled backstepping,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 4, pp. 1439–1446, 2004. [89] S. S. Ge and J. Wang, “Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients,” IEEE Transactions on Automatic Control, vol. 48, no. 8, pp. 1463–1469, 2003. [90] S. S. Ge and C. Wang, “Adaptive neural control of uncertain mimo nonlinear systems,” IEEE Transactions on Neural Networks, vol. 15, no. 3, pp. 674–692, 2004. [91] S. S. Ge, C. C. Hang, and T. Zhang, “Nonlinear adaptive control using neural network and its application to CSTR systems,” Journal of Process Control, vol. 9, no. 4, pp. 313–323, 1999. [92] G. Lightbody and G. W. Irwin, “Direct neural model reference adaptive control,” IEE Proceedings Control Theory & Applications, vol. 142, no. 1, pp. 31–43, 1995. 155 Bibliography [93] W. Lin and C. Qian, “Adaptive control of nonlinearly parameterized systems: the smooth feedback case,” IEEE Transactions on Automatic Control, vol. 47, no. 8, pp. 1249–1266, 2002. [94] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York: Wiley, 1994. [95] S. S. Ge, “Advanced control techniques of robotic manipulators,” in Proceedings of the American Control Conference, Philadelphia, Pennsylvania, Jun 1998, pp. 2185–2199. [96] S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators. London: World Scientific, 1998. [97] C. M. Kwan, F. L. Lewis, and Y. H. Kim, “Robust neural network control of flexible-joint robots,” in Proceedings of the 34th Conference on Decision & Control, New Orleans, Louisiana, Dec 1995, pp. 1296–1301. [98] Ø. N. Smogeli, A. J. Sørensen, and T. I. Fossen, “Design of a hybrid power/torque thruster controller with thrust loss estimation,” in Proceedings of the IFAC Conference on Control Applications in Marine Systems, Ancona, Italy, Jul 2004, pp. 1443–1447. [99] R. Skjetne, T. I. Fossen, and P. V. Kokotovi´c, “Adaptive maneuvering, with experiments, for a model ship in a marine control laboratory,” Automatica, vol. 41, pp. 289–298, 2005. [100] J. E. Slotine and W. Li, Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [101] L. L. Whitcomb and D. R. Yoerger, “Development, comparison, and preliminary experimental validation of nonlinear dynamic thruster models,” IEEE Journal of Oceanic Engineering, vol. 24, no. 4, pp. 481–494, 1999. [102] ——, “Development, comparison, and preliminary experimental validation of nonlinear dynamic thruster models,” IEEE Journal of Oceanic Engineering, vol. 24, no. 4, pp. 495–506, 1999. [103] J. S. Wu and M. Hsieh, “Torsional vibration of a damped shaft system using the analytical-and-numerical-combined method,” Marine Technology, vol. 38, no. 4, pp. 250–260, 2001. 156 Appendix A Derivations for Simulation The derivations of the partial derivatives that are required for the design of an adaptive NN control using decoupled backstepping in the simulation conducted in Section 5.5.1 is presented below. In the subsequent derivations, the following notations are adopted to make the presentation more concise. ∂a ˆ˙ ∂a ˆ˙ ∂a ˙ ∂a ˆ˙ ∂a ˙ W1 + W2 + W + θ1 + θ2 + ˆ1 ˆ2 ˆ3 ∂θ1 ∂θ2 ∂W ∂W ∂W ∂a := ∂ θ˙1 ∂a := ∂ θ˙2 ζa := νa a ∂a (i+1) θ (i) d i=0 ∂θd (i) where θd denotes the ith time derivative of θd . z1 = = = z˙1 = ζz1 = νz = ζz˙ = ζζz1 νζz1 ν˙z1 θ˙1 − α0 θ˙1 − (−c0,1 (θ1 − θd ) + θ˙d ) c0,1 θ1 + θ˙1 − c0,1 θd − θ˙d ζz1 + νz1 θ¨1 c0,1 θ˙1 − c0,1 θ˙d − θ¨d ζζz1 + νζz1 θ¨1 (3) = −c0,1 θ¨d − θd = c0,1 = (3) (4) ζζζz = −c0,1 θd − θd νζζz = ζνζz = ννζz = In the above derivations, the terms ζz1 and νz1 are required in computing the partial 157 derivatives of αi for i = 1, 2, while the terms ζζz1 , νζz1 , ζνz1 and ννz1 are required in computing the partial derivatives of α2 and α3 while the rest of the terms are required in computing the partial derivatives of α3 . From the above definitions and the definition of φi in (5.21) and (5.21), we have φi = ζαi φ0 = ζα0 = −c0,1 (θ˙1 − θ˙d ) + θ¨d φ˙0 = ζφ0 + νφ0 θ¨1 (3) ζφ = c0,1 θ¨d + θ d νφ0 = −c0,1 ζφ˙ = ζζφ0 + νζφ0 θ¨1 (3) (4) (4) (5) ζζφ0 = c0,1 θd + θd ζζζφ = c0,1 θd + θd νφ˙ = S1 = [ θ1 θ˙1 z1 φ0 T , , , ] S˙1 ζS1 νS1 ζS˙ = = = = ζS1 + νS1 θ¨1 θ˙1 K1,1 S1 + ζz1 K1,3 S1 + ζφ0 K1,4 S1 K1,2 S1 + νz1 K1,3 S1 + νφ0 K1,4 S1 ζζS1 + νζS1 θ¨1 ζζS1 = (θ˙1 K1,5 S1 + θ˙1 K1,1 ζS1 ) + (ζζz1 K1,3 S1 + ζz21 K1,7 S1 + ζz1 K1,3 ζS1 ) νζS1 νS˙ ζνS1 +(ζζφ0 K1,4 S1 + ζφ20 K1,8 S1 + ζφ0 K1,4 ζS1 ) = (K1,1 S1 + θ˙1 K1,1 νS1 ) + (νζz1 K1,3 S1 + ζz1 νz1 K1,7 S1 + ζz1 K1,3 νS1 ) +(ζφ0 νφ0 K1,8 S1 + ζφ0 K1,4 νS1 ) = ζνS1 + ννS1 θ¨1 = K1,2 ζS1 + (νz1 ζz1 K1,7 S1 + νz1 K1,3 ζS1 ) + νφ0 ζφ0 K1,8 S1 + νφ0 K1,4 ζS1 ννS1 = (K1,6 S1 + K1,2 νS1 ) + (νz21 K1,7 S1 + νz1 K1,3 νS1 ) + (νφ20 K1,8 S1 +νφ0 K1,4 νS1 ) ˙ ζζS1 = ζζζS + νζζS θ¨1 ζζζS 2 = θ˙1 K1,5 ζS1 + (θ˙1 K1,5 ζS1 + θ˙1 K1,1 ζζS1 ) + (ζζζz K1,3 S1 + ζζz1 ζz1 K1,7 S1 +ζζz1 K1,3 ζS1 ) + (2ζz1 ζζz1 K1,7 S1 + ζz21 K1,7 ζS1 ) + (ζζz1 K1,3 ζS1 +ζz21 K1,7 ζS1 + ζz1 K1,3 ζζS1 ) + (ζζζφ K1,4 S1 + ζζφ0 ζφ0 K1,8 S1 +ζζφ0 K1,4 ζS1 ) + (2ζφ0 ζζφ0 K1,8 S1 + ζφ20 K1,8 ζS1 ) + (ζζφ0 K1,4 ζS1 +ζφ20 K1,8 ζS1 + ζφ0 K1,4 ζζS1 ) 158 νζζS = (2θ˙1 K1,5 S1 + θ˙1 K1,5 νS1 ) + (K1,1 ζS1 + θ˙1 K1,1 νζS1 ) + (νζζz K1,3 S1 +ζζz1 νz1 K1,7 S1 + ζζz1 K1,3 νS1 ) + (2ζz1 νζz1 K1,7 S1 + νζ˙S1 ζνζS ζz21 K1,7 νS1 ) +(νζz1 K1,3 ζS1 + ζz1 νz1 K1,7 ζS1 + ζz1 K1,3 νζS1 ) + ζφ20 K1,8 νS1 +(ζζφ0 νφ0 K1,8 S1 + ζζφ0 K1,4 νS1 ) + (ζφ0 νφ0 K1,8 ζS1 + ζφ0 K1,4 νζS1 ) = ζνζS + ννζS θ¨1 = (θ˙1 K1,5 S1 + K1,1 ζS1 ) + (θ˙1 K1,5 νS1 + θ˙1 K1,1 ζνS1 ) + (νζz1 ζz1 K1,7 S1 +νζz1 K1,3 ζS1 ) + (ζζz1 νz1 K1,7 S1 + ζz1 νz1 K1,7 ζS1 ) + (ζζz1 K1,3 νS1 +ζz21 K1,7 νS1 + ζz1 K1,3 ζνS1 ) + (ζζφ0 νφ0 K1,8 S1 + ζφ0 νφ0 K1,8 ζS1 ) +(ζφ20 K1,8 νS1 + ζφ0 K1,4 ζνS1 ) = K1,1 νS1 + (K1,1 νS1 + θ˙1 K1,1 ννS1 ) + (νζz1 νz1 K1,7 S1 + ζz1 K1,3 νS1 ) ζν˙S1 +(νζz1 νz1 K1,7 S1 + ζz1 νz1 K1,7 νS1 ) + (νζz1 K1,3 νS1 + ζz1 νz1 K1,7 νS1 +ζz1 K1,3 ννS1 ) + ζφ0 νφ0 K1,8 νS1 + (ζφ0 νφ0 K1,8 νS1 + ζφ0 K1,4 ννS1 ) = ζζνS + νζνS θ¨1 ννζS ζζνS = K1,2 ζζS1 + (νz1 ζζz1 K1,7 S1 + νz1 ζz1 K1,7 ζS1 ) + (νz1 ζz1 K1,7 ζS1 +νz1 K1,3 ζζS1 ) + (νφ0 ζζφ0 K1,8 S1 + νφ0 ζφ0 K1,8 ζS1 ) + (νφ0 ζφ0 K1,8 ζS1 +νφ0 K1,4 ζζS1 ) νζνS = (K1,6 ζS1 + K1,2 νζS1 ) + (νz1 νζz1 K1,7 S1 + νz1 ζz1 K1,7 νS1 ) + (νz21 K1,7 νS1 νν˙S1 +νz1 K1,3 ννS1 ) + (νφ0 νζφ0 K1,8 S1 + νφ0 ζφ0 K1,8 νS1 ) + (νφ20 K1,8 ζS1 +νφ0 K1,4 νζS1 ) = ζννS + νννS θ¨1 ζννS = K1,6 ζS1 + K1,2 ζνS1 + νz21 K1,7 ζS1 + (νz1 ζz1 K1,7 νS1 + K1,3 ζνS1 ) +νφ20 K1,8 ζS1 + (νφ0 ζφ0 K1,8 νS1 + νφ0 K1,4 ζνS1 ) νννS = K1,6 νS1 + (K1,6 νS1 + K1,2 ννS1 ) + νz21 K1,7 νS1 + (νz21 K1,7 νS1 +K1,3 ννS1 ) + νφ20 K1,8 νS1 + (νφ20 K1,8 νS1 + νφ0 K1,8 ννS1 ) where K1,1 = diag[− 2( θ11 − µ111 ) λ1,1 , .,− 2( θ11 − µ11l1 ) , .,− 2( θ21 − µ12l1 ) , .,− 2( z51 − µ13l1 ) ˙ K1,2 = diag[− K1,3 = diag[− 2( θ21 − µ121 ) 2 λ1,1 2( z51 − µ131 ) λ1,1 159 λ1,l1 ] ˙ 2 λ1,l1 λ1,l1 ] ] K1,4 = diag[− 2( φ60 − µ141 ) λ1,1 K1,5 = diag[− 2 λ1,1 K1,6 = diag[− 2 λ1,1 K1,7 = diag[− 2 λ1,1 K1,8 = diag[− 2 λ1,1 , .,− , .,− , .,− , .,− , .,− 2 λ1,l1 2 2 λ1,l1 2 λ1,l1 2 λ1,l1 2( φ60 − µ14l1 ) λ1,l1 ] ] ] ] ] ˆ˙ = −Γ1 [S1 z1 + σ1 W ˆ1 ] W ¨ˆ˙ ¨ W = ζW ˆ˙ + νW ˆ˙ θ1 ζWˆ˙ ˆ˙ ] = −Γ1 [(ζS1 z1 + S1 ζz1 ) + σ1 W νWˆ˙ = −Γ1 [(νS1 z1 + S1 νz1 ) + 0] 1 ¨˙ = ζζ + νζ θ¨1 ζW ˆ˙ ˆ˙ ˆ W W 1 ζζ ˆ˙ = −Γ1 [(ζζS1 z1 + ζS1 ζz1 ) + (ζS1 ζz1 + S1 ζζz1 ) + σ1 ζWˆ˙ ] W1 νζ ˆ˙ W = −Γ1 [(νζS1 z1 + ζS1 νz1 ) + (νS1 ζz1 + S1 νζz1 ) + σ1 νWˆ˙ ] θ¨1 νW ¨˙ = ζν ˆ˙ + νν ˆ˙ ˆ W1 W1 ζν ˆ˙ = −Γ1 [(ζνS1 z1 + νS1 ζz1 ) + (ζS1 νz1 + 0)] W1 νν ˆ˙ W = −Γ1 [(ννS1 z1 + νS1 νz1 ) + (νS1 νz1 + 0)] α1 = −c1,1 z1 + Wˆ1T S1 + θ1 α˙ = ζα1 + να1 θ¨1 ˙ ζα1 = −c1,1 ζz1 + Wˆ1T S1 + Wˆ1T ζS1 + θ˙1 ν = −c + WˆT ν α1 1,1 S1 ζ˙α1 = ζζα1 + νζα1 θ¨1 ζζα1 = −c1,1 ζζz1 + (ζ ˙ S1 Wˆ1T νζα1 = −c1,1 νζz1 + (ν ˙ S1 Wˆ1T ˙ ˙ + Wˆ1T ζS1 ) + (Wˆ1T ζS1 + Wˆ1T ζζS1 ) ˙ + Wˆ1T νS1 ) + Wˆ1T νζS1 + 160 ν˙ α1 = ζνα1 + ννα1 θ¨1 ˙ ζνα1 = Wˆ1T νS1 + Wˆ1T ζνS1 ν = WˆT ν ν α1 νS1 ζ˙ζα1 = ζζζα + νζζα θ¨1 ζζζα = −c1,1 ζζζz + (ζζ ˆ˙ W1T ˙ ζS1 Wˆ1T = + (νζ ˆ˙ W1T ˙ ζS1 Wˆ1T + (ζ ˙ + Wˆ1T ζζS1 ) ˙ ˙ + Wˆ1T ζζS1 ) + (Wˆ1T ζζS1 + Wˆ1T ζζζS ) +(ζ νζζα ˙ ζS1 ) Wˆ1T S1 + ζ S1 + ζ ˙ νS1 ) Wˆ1T + (ν ˙ ζS1 ) Wˆ1T + (ζ ˙ νS1 ) Wˆ1T + (ν ˙ ζS1 Wˆ1T ˙ + Wˆ1T νζS1 ) + (ν ˙ ζS1 Wˆ1T ˙ + Wˆ1T νζS1 ) +Wˆ1T νζζS ν˙ ζα1 = ζνζα + ννζα θ¨1 ζνζα ννζα = + (ζν ˆ˙ W1T = + (νν ˆ˙ W1T S1 + ν S1 + ν ˙ νS1 Wˆ1T ˙ νS1 Wˆ1T ˙ ˙ + Wˆ1T ζνS1 )) + (Wˆ1T νζS1 + Wˆ1T ζνζS ) ˙ + Wˆ1T ννS1 ) + Wˆ1T ννζS ζ˙να1 = ζζνα1 + νζνα1 θ¨1 ˙ ˙ ζζνα1 = (ζ ˆ˙ T νS1 + Wˆ1T ζνS1 ) + (Wˆ1T ζνS1 + Wˆ1T ζζνS ) W1 νζνα1 = (ν ˙ νS1 Wˆ1T ˙ + Wˆ1T ννS1 ) + Wˆ1T νζνS ν˙ να1 = ζννα1 + νννα1 θ¨1 ˙ ζννα1 = Wˆ1T ννS1 + Wˆ1T ζννS = Wˆ1T νννS νννα1 S2 = [ θ1 θ˙1 θ2 να1 z2 T , , , , ] S˙ ζS2 νS2 ζ˙S = = = = ζS2 + νS2 θ¨1 θ˙1 K2,1 S2 + θ˙2 K2,3 S2 + ζνα1 K2,4 S2 + ζz2 K2,5 S2 K2,2 S2 + ννα1 K2,4 S2 + νz2 K2,5 S2 ζζ + νζ θ¨1 + ϕζ θ¨2 S2 S2 S2 ζζS2 = (θ˙1 K2,6 S2 + θ˙1 K2,1 ζS2 ) + (θ˙2 K2,8 S2 + θ˙2 K2,3 ζS2 ) + (ζζνα1 K2,4 S2 +ζν2α1 K2,9 S2 + ζνα1 K2,4 ζS2 ) + (ζζz2 K2,5 S2 + ζz22 K2,10 S2 + ζz2 K2,5 ζS2 ) 161 νζS2 = (K2,1 S2 + θ˙1 K2,1 νS2 ) + θ˙2 K2,3 νS2 + (νζνα1 K2,4 S2 + ζνα1 ννα1 K2,8 S2 ν˙ S2 ζνS2 +ζνα1 K2,4 νS2 ) + (νζz2 K2,5 S2 + ζz2 νz2 K2,10 S2 + ζz2 K2,5 νS2 ) = K2,3 S2 = ζνS2 + ννS2 θ¨1 + ϕνS2 θ¨2 = K2,2 ζS2 + (ζννα1 K2,4 S2 + ννα1 ζνα1 K2,9 S2 + ννα1 K2,4 ζS2 ) + (ζνz2 K2,5 S2 ννS2 +νz2 ζz2 K2,10 S2 + νz2 K2,5 ζS2 ) = (K2,7 S2 + K2,2 νS2 ) + (νννα1 K2,4 S2 + νν2α1 K2,9 S2 + ννα1 K2,4 νS2 ) ϕζS2 +(ννz2 K2,5 S2 + νz22 K2,10 S2 + νz2 K2,5 νS2 ) where K2,1 = diag[− 2( θ11 − µ211 ) λ2,1 , .,− 2( θ11 − µ21l1 ) , .,− 2( θ21 − µ21l1 ) , .,− 2( θ32 − µ21l1 ) ˙ K2,2 = diag[− K2,3 = diag[− K2,4 = diag[− K2,5 = diag[− K2,6 = diag[− K2,7 = diag[− 2( θ21 − µ221 ) 2 λ2,1 2( θ32 − µ231 ) λ2,1 λ2,1 λ2,1 2 λ2,1 2 2 λ2,1 , .,− λ2,l1 , .,− , .,− , .,− 2 λ2,l1 2 λ2,l1 2 2 λ2,l1 K2,10 ] λ2,l1 2( z82 − µ21l1 ) λ2,l1 ] ] 2 , . . . , − ] 2 2 λ2,1 λ2,l1 2 = diag[− 2 , . . . , − 2 ] λ2,1 λ2,l1 2 = diag[− 2 , . . . , − 2 ] λ2,1 λ2,l1 ˆ˙ = −Γ2 [S2 z2 + σ2 W ˆ2 ] W ˆ¨2 = ζW2 + νW2 θ¨1 + ϕW2 θ¨2 W ˆ˙ ] ζW2 = −Γ2 [(ζS2 z2 + S2 ζz2 ) + σ2 W νW2 = −Γ2 [(νS2 z2 + S2 νz2 ) + 0] ϕW2 = −Γ2 [ϕS2 z2 + S2 ϕz2 ) + 0] 162 ] 2( 17 να1 − µ21l1 ) K2,8 = diag[− K2,9 ] ˙ 2( 17 να1 − µ241 ) 2( z82 − µ251 ) λ2,l1 ] ] α2 = −c2,1 z2 + Wˆ2T S2 + φ1 α˙ = ζα2 + να2 θ¨1 ˙ ζα2 = −c2,1 ζz2 + Wˆ2T S2 + Wˆ2T ζS2 + ζφ1 ν = −c ν + WˆT ν + ν α2 2,1 z2 S2 φ1 ζ˙α2 = ζζα2 + νζα2 θ¨1 ˙ ˙ + Wˆ2T ζS2 ) + (Wˆ2T ζS2 + Wˆ2T ζζS2 ) + ζζφ1 ζζα2 = −c2,1 ζζz2 + (ζ ˙ S2 Wˆ2T νζα2 = −c2,1 νζz2 + (ν ˙ + Wˆ2T ν(S2 ) ) + Wˆ2T νζS2 + νζφ1 ˙ S2 Wˆ2T ϕζα2 = −c2,1 ϕζz2 + (ϕ ˙ S2 Wˆ2T + 0) + Wˆ2T ϕζS2 ν˙ α2 = ζνα2 + ννα2 θ¨1 ˙ ζνα2 = −c2,1 ζνz2 + Wˆ2T νS2 + Wˆ2T ζνS2 + ζνφ1 ννα2 = −c2,1 ννz2 + Wˆ2T ννS2 + ννφ1 S3 = [ θ1 θ˙1 θ2 θ˙2 να2 z3 φ2 T , , , , , , ] , S˙3 ζS3 νS3 ϕS3 = = = = 10 11 ζS3 + νS3 θ¨1 + ϕS3 θ¨2 θ˙1 K3,1 S3 + θ˙2 K3,3 S3 + ζνα3 K3,5 S3 + ζz3 K3,6 S3 + ζφ2 K3,7 S3 K3,2 S3 + ννα3 K3,5 S3 + νz3 K3,6 S3 + νφ2 K3,7 S3 K2,4 S3 + K3,6 S3 + ϕφ2 K3,7 S3 where K3,1 = diag[− 2( θ11 − µ311 ) λ3,1 , .,− 2( θ11 − µ31l1 ) , .,− 2( θ21 − µ32l1 ) , .,− 2( θ32 − µ33l1 ) , .,− 2( θ42 − µ34l1 ) ˙ K3,2 = diag[− K3,3 = diag[− 2( θ21 − µ321 ) 2 λ3,1 2( θ32 − µ331 ) λ3,1 K3,4 = diag[− K3,5 = diag[− λ3,1 2( ν α3 163 2 λ3,l1 λ3,l1 ] ] ˙ − µ351 ) λ3,1 ] ˙ ˙ 2( θ42 − µ341 ) λ3,l1 , .,− 2( λ3,l1 ν α3 ] − µ35l1 ) λ3,l1 ] K3,6 = diag[− K3,7 = diag[− 2( z103 − µ361 ) 0λ3,1 2( φ112 − µ371 ) 1λ3,1 , .,− 2( z103 − µ36l1 ) , .,− 2( φ112 − µ37l1 ) 0λ3,l1 1λ3,l1 α3 = −c3,1 z3 + Wˆ3T S3 + φ2 α˙3 = ζα3 + να3 θ¨1 + ϕα3 θ¨2 ˙ ζα3 = −c3,1 ζz3 + (Wˆ3T S3 + Wˆ3T ζS3 ) + ζφ2 ν = −c ν + WˆT ν + ν α3 ϕα3 3,1 z3 S3 φ2 = −c3,1 ϕz3 + Wˆ3T ϕS3 + ϕφ2 164 ] ] Appendix B Author’s Publications Journal Papers: [1] Z. Li, P. Y. Tao, S. S. Ge ,M. Adams, and W. S. Wijesoma, “Robust Adaptive Control of Cooperating Mobile Manipulators with Relative Motion,” IEEE Transactions on Systems, Man, and Cybernetics–Part B: Cybernetics, Accepted. [2] P. Y. Tao, S. S. Ge, T. H. Lee, X. Q. Chen, and W. Lin, “Control of Marine Shafting System Using a Class of Feedforward Approximators,” IEEE Transactions on Control Systems Technology, Submitted. [3] P. Y. Tao, S. S. Ge, and T. H. Lee, “Adaptive Neural Network Control for Marine Shafting System Using Dynamic Surface Control,” IEEE Journal of Oceanic Engineering, Submitted. Conference Papers: [1] P. Y. Tao, S. S. Ge, T.H. Lee, and X.Q. Chen, “Robust Adaptive Control of Uncertain Nonholonomic Systems using Domination Design,” Proceedings of the SICE Annual Conference, pp. 2472-2477, Okayama, Japan, Aug, 2005. [2] P. Y. Tao, S. S. Ge, T.H. Lee, and X.Q. Chen, “Thruster and Vibration Control of Marine Powertrain Using a Class of Feedforward Approximators,” Proceedings of the IEEE Conference on Control Applications, pp. 25892594, Munich, Germany, Oct 2006. [3] P. Y. Tao, S. S. Ge, and T.H. Lee, “Adaptive Neural Network Control for Marine Shafting System Using Dynamic Surface Control,” Proceedings of the IEEE Conference on Control Applications, pp. 717-722, Singapore, Oct 2007. 165 [...]...1.1 Adaptive Variable Structure Control 1.1 Adaptive Variable Structure Control Adaptive control has been the subject of much research for control theorists over the past half a century and under the efforts of many researchers, adaptive control has been extended for larger classes of nonlinear systems and issues such as overparametrization and robustness of the closed-loop system has been dealt... dynamic surface control DSC was meant as a dynamic extension of the MSS control where the derivative of the virtual controls are obtained using numerical differentiation compared with dynamic filters in DSC and hence the name dynamic surface control Although some of the subsequent papers extending the work in [65] do not use variable structure control in the control design, the technique of using filters... stability of the closed-loop system since the control will be saturated In [27], adaptive output feedback control was developed for a class of nonlinear systems in output feedback form In the early developments of adaptive nonlinear control, the uncertain systems are assumed to be linearly parametrization, where the unknown parameters appear linearly in the system and the regressors are known Adaptive control. .. uncertain nonlinear strict-feedback systems have been investigated [44, 45, 49, 54–57] The systems 3 1.1 Adaptive Variable Structure Control considered are generally affine systems with either known or unknown virtual control coefficients For the unknown case, the controller singularity problem was addressed in [49, 54–57] Neural network control of nonlinear strict-feedback systems is well documented in the... recent years, control of non-affine nonlinear systems have captured the attention of researchers and poses a challenge to control theorists The main impediment in solving this control problem directly is that even if the inverse is known to exist, it may be impossible to construct it analytically Consequently, no control system design is possible along the lines of conventional model based control Fundamental... systems For such systems, the main difficulty is in dealing with non-affine functions, particularly in the final step of backstepping, where circular argument of control may appear In this thesis, an Adaptive Variable Structure Control (AVSC) is presented, which combines the ideas of Variable Structure Control (VSC), Mean Value Theorem, neural network parametrization to solve the control problem of non-affine... absolute value of the output tracking error Traditional adaptive control was initially targeted at linear systems and subsequently extended to nonlinear systems Motivated by the results in feedback linearization techniques [17], adaptive control was developed for a class of nonlinear systems [18–21] Robustness issues associated with adaptive nonlinear control has also been studied for systems with bounded... filters to approximate the virtual controls is now termed as dynamic surface control [66, 69, 70] Dynamic surface control for a class of uncertain systems with the uncertainties 5 1.2 Adaptive NN Control of Marine Shafting System bounded by a known function is developed in [65] The incorporation of Radial Basis Functions (RBF) Neural Networks (NN) in dynamic surface control is presented in [66] Simulation... appearing in the time derivatives of the virtual controls In the decoupled backstepping approach, due to the structure of the dynamic model considered and the approach adopted, the NNs in every alternate step of the backstepping are required exclusively for the approximation of the time derivatives of the virtual controls In contrast, the time derivatives of the virtual controls are approximated by first-order... dynamic surface control, thus removing the need of NNs for every alternate step in the backstepping which effectively reduces the number of NNs required in the system by half Therefore, through the application of dynamic surface control technique, it is possible to reduce the number of NNs required as well as reducing the size of the NNs in the control 1.3 Thesis Outline The remainder of this thesis . increasingly complex systems have led to extensive research in the field of control theory to develop novel control methodologies or to extend existing control designs for a larger class of systems and. the virtual controls is now termed as dynamic surface control [66, 69, 70]. Dynamic surface control for a class of uncertain systems with the uncertainties 5 1.2. Adaptive NN Control of Marine. the design of the AVSC with dynamic iii surface control (DSC) is explored to further reduce the size of the NNs required. In the subsequent chapter, the application of intelligent control is