IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 5, MAY 2000 949 stochastic stability framework, stability of all operation modes is not even required. REFERENCES [1] W. P. Blair Jr. and D. D. Sworder, “Continuous-time regulation of a class of econometric models,” IEEE Trans. Syst. Man Cybern., vol. 5, pp. 341–346, 1975. [2] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM, 1994. [3] P. Colaneri, J. C. Geromel, and A. Locatelli, Control Theory and De- sign—An / Viewpoint. New York: Academic, 1997. [4] O. L. V. Costa, J. B. R. do Val, and J. C. Geromel, “Continuous-time state-feedback -control of Markovian jump linear systems via convex analysis,” Automatica, vol. 35, no. 2, pp. 259–268, 1999. [5] D. P. de Farias, “Otimização e controle de sistemas com parâmetros su- jeitos a saltos markovianos,” M.Sc. thesis (in Portuguese), UNICAMP, 1998. [6] C. E. de Souza and M. D. 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[12] M. Mariton and P. Bertrand, “Output feedback for a class of linear sys- tems with jump parameters,” IEEE Trans. Automat. Contr., vol. 30, pp. 898–900, 1985. [13] M. Mariton, “Almost sure and moments stability of jump linear sys- tems,” Syst. Contr. Lett., vol. 11, pp. 393–397, 1988. [14] , Jump Linear Systems in Automatic Control. New York: Marcel Dekker, 1990. [15] K. S. Narendra and S. S. Tripathi, “Identification and optimization of aircraft dynamics,” J. Aircraft, vol. 10, no. 4, pp. 193–199, 1973. [16] M. A. Rami and L. El Ghaoui, “LMI optimization for nonstandard Riccati equations arising in stochastic control,” IEEE Trans. Automat. Contr., vol. 41, pp. 1666–1671, 1996. Nonlinear Repetitive Control Jayati Ghosh and Brad Paden Abstract—Repetitive controllers are generally applied to reject periodic disturbances and to track periodic reference signals with a known period. Their design is based on The Internal Model Principle, proposed by Francis and Wonham. This paper describes a new finite-dimensional SISO repetitive controller for two different classes of nonlinear plants. Simulation results show asymptotic tracking of the periodic reference signal by the proposed repetitive controller in closed loop up to the th harmonic frequency. A proof of robustness of the repetitive control system to small nonlinearities, like actuator nonlinearities, is provided. Index Terms—Internal model principle, nonlinear, tracking. I. INTRODUCTION Periodic signals commonly occur in robotics, servo mechanisms, and other similar tracking scenarios, either in form of reference inputs or disturbances. In linear time-invariant plants, repetitive control builds on the well-known internal model principle [1]–[3] to provide exact asymptotic output tracking of periodic inputs. The internal model prin- ciple states that the output of a plant can be made to asymptotically track a class of reference commands without a steady-state error if the generator for the reference signal is included in the stable closed-loop system. If a periodic input signal has a finite Fourier series, then a finite number of internal models (one for each harmonic) can be used to pro- duce asymptotic tracking. Likewise, if the periodic input has an infinite Fourier series, an infinite number of controller models (i.e., -axis poles) are required for exact tracking. Fortunately, a simple delay line can be used to produce an infinite number of poles, but the system dy- namics are, nonetheless, infinite dimensional. Such periodic tracking problems are surprisingly common. For example, every computer disk drive uses some type of linear repetitive control to compensate for repeatable runout in the disk bearings. Other applications of signifi- cant economic value include eccentricity compensation in rolling mills, noncircular machining of pistons and camshafts, AFM control, and op- tical turning. The innovation of repetitive control was motivated by a power supply regulation problem and is due to Inoue et al. [4]. Early progress was made in papers by Nakano, Iwai, Omata, and Hara [5], [6], culminating in a seminal paper on the stability of linear infinite-dimensional repet- itive controllers [7]. Repetitive control theory became more accessible with the appealing discrete-time formulation of Tomizuka et al. [8]; the discrete-time formulation was developed further to cover robustness analysis [9]. Disturbance rejection is a particularly important problem in repetitive control, and this has been addressed in the context of the discrete-time formulation for disk-drive shock disturbance rejection [10]. The discrete-time formulation also allows segments of a periodic reference input to be selected for exact tracking, thus saving computer memory [11]. The repetitive control theory community joined forces with the multivariable control community beginning with the work of Ledwich and Bolton [12]. Further research on linear repetitive control design is due to Güvenc [13] where sensitivity minimization at dis- crete points is used to design a repetitive controller. Roh and Chung Manuscript received May 26, 1998; revised April 1, 1999 and July 28, 1999. Recommended by Associate Editor, J. Si. This work was supported in part by the National Science Foundation under Grant CMS-9800294. The authors are with the Department of Mechanical and Environmental En- gineering, University of California, Santa Barbara, CA 93106 USA (e-mail: jayati@engineering.ucsb.edu; paden@engineering.ucsb.edu). Publisher Item Identifier S 0018-9286(00)04061-7. 0018–9286/00$10.00 © 2000 IEEE