未知相对度非线性系统的一阶D型迭代学习控制

1、 560 4 ACTA AUTOMATICA SINICA Vol. 31 Simulation example Consider a SISO nonlinear system x 1 (t = x 2 (t x 2 (t = 4.0(x1 (t3 4.0x2 (t + u(t y (t = 2.0x1 (t The relative degree of the nonlinear system is r = 2. The desired trajectory is taken as yd (t = sin(2t(0 t 1. The parameters of the series-wou

2、nd subsystem and the shunt-wound subsystem are taken as = = 50000. The desired initial states of the nonlinear system are supposed to be all zero. The learning gain is taken as P = 0.95, and the control input at the rst iteration is taken as u1 (t = yd (t. The iterative learning control is operated

3、for 50 iterations. The supremum of the tracking error is shown in Fig. 2. The output trajectory and the tracking error of the actual controlled system at the 50th iteration are shown in Fig. 3 and Fig. 4, respectively. Fig. 2 The supremum of the tracking error Fig. 3 The output trajectory of the act

4、ual controlled system at the 50th iteration Fig. 4 The tracking error of the actual controlled system at the 50th iteration The relative degree of the nonlinear system is r = 2, a second-order dierential operation on the output is required for the classical D-type ILC law, and it is possible to desi

5、gn a convergent ILC law only when the sign and bound of the gain of relative degree of the controlled system are exactly known. And more important, when neither the relative degree nor the gain of relative degree of the controlled nonlinear system is known, the classical D-type ILC is helpless. A rs

6、t-order dierential operation is only required for the scheme based on the dummy model presented in this paper, and the detailed knowledge of the sign and bound of the gain of relative degree of the controlled system is avoided. No. 4 SONG Zhao-Qing et al.: First-order D-type Iterative Learning Contr

7、ol for Nonlinear · · · 561 5 Conclusion In solving the problems of the classical D-type ILC law depending on the relative degree of the controlled system and the gain of relative degree, and of the high-order dierential operation, a novel design method of rst-order D-type ILC based on

8、 the dummy model is presented. The main idea of the method is constructing a dummy model with relative degree one and with gain of relative degree that can be designed arbitrarily by connecting in series and connecting in parallel with a rst-order subsystem with the controlled system. The meaning of

9、 the dummy model here is that the shunt-wound subsystem is not included in the control process, it is only a dummy part when calculating the tracking error and its derivative. The presented scheme does not need the information of the relative degree and the gain of relative degree of the controlled

10、system, not does it rely on the parameter of the controlled system. A low order D-type ILC law, or in a precise word, a rst-order D-type ILC law, can be designed for a class of systems with high order relative degree. The rst-order partial dierential of the output with respect to the state vector is

11、 only required, the property of continuous dierentiability is not required for the all vector elds of the controlled nonlinear systems. The proof of the theorem and the simulation example demonstrate the feasibility and eectiveness of the presented scheme. References 1 Sun Ming-Xuan, Huang Bao-Jian.

12、 Iterative Learning Control. Beijing, China: National Defence Industrial Press, 1999 2 Sugie T, Ono T. An iterative learning control law for dynamical systems. Automatica, 1991, 27(4: 729732 3 Porter B, Mohamed S S. Iterative learning control of partially irregular multivariable plants with initial

13、impulsive action. International Journal of Systems Science, 1991, 22(3: 447454 4 Ahn H S, Choi C H, Kim K B. Iterative learning control for a class of nonlinear systems. Automatica, 1993, 29(6: 15751578 5 Sun M, Wang D, Xu G. Initial shift problem and its ILC solution for nonlinear systems with high

14、er relative degree. In: Proceeding of the American Control Conference, Chicago, Illinois, USA: Prentice Hall, 2000. 2274231 6 Khalil H K. Nonlinear Systems (Third Edition. Upper Saddle River: PTR Prentice-Hall, 2002 7 Sun M, Wang D, Xu G. Sampled-data iterative learning control for SISO nonlinear sy

15、stems with arbitrary relative degree. In: Proceeding of the American Control Conference, Chicago, Illinois, USA: 2000. 667671 8 Wang D. On D-type and P-type ILC designs and anticipatory approach. International Journal of Control, 2000, 73(10: 890901 9 Sun M, Wang D. Sampled-data iterative learning c

16、ontrol for nonlinear systems with arbitrary relative degree, Automatica, 2001, 37(2: 283289 10 Sun M, Wang D. Anticipatory Iterative Learning Control for Nonlinear Systems with Arbitrary Relative Degree. IEEE Transactions on Automatic Control, 2001, 46(5: 783788 11 Ioannou P A, Sun J. Robust Adaptiv

17、e Control. Upper Saddle River: PTR Prentice-Hall, 1996 SONG Zhao-Qing Received his bachelor in July 1992 from the Antiaircraft Academy, and master of control theory and control engineering in 1998 from the Naval Aeronautical Engineering Academy. From 1998, he jointed the faculty of the Department of

18、 Automatic Control of the Naval Aeronautical Engineering Academy, where he is now an associate professor. Since 2001, he has been a Ph. D. candidate of control theory and control engineering in Beijing University of Aeronautics & Astronautics. His current research interests include system identi

19、cation, iterative learning control, neural network, and adaptive control. MAO Jian-Qin Received her bachelor degree in mathematics and mechanics from Beijing University in 1964, and master and Ph. D. degrees in aeronautical navigation and control engineering from Beijing University of Aeronautics and Astronautics