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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 10, OCTOBER 2000 1893 Dynamic Surface Control for a Class of Nonlinear Systems D. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes Abstract— A new method is proposed for designing controllers with ar- bitrarily small tracking error for uncertain, mismatched nonlinear systems in the strict feedback form. This method is another “synthetic input tech- nique,” similar to backstepping and multiple surface control methods, but with an important addition, I low pass filters are included in the design where is the relative degree of the output to be controlled. It is shown that these low pass filters allow a design where the model is not differentiated, thus ending the complexity arising due to the “explosion of terms” that has made other methods difficult to implement in practice. The backstep- ping approach, while suffering from the problem of “explosion of terms” guarantees boundedness of tracking errors globally; however, the proposed approach, while being simpler to implement, can only guarantee bounded- ness of tracking error semiglobally, when the nonlinearities in the system are non-Lipschitz. Index Terms— Integrator backstepping, nonlinear control system design, semiglobal tracking, sliding mode control, strict feedback form. I. INTRODUCTION Tremendous strides have been made in the past 25 years in the area of controller design for nonlinear systems. Variable structure control or sliding mode control [5], [25], uses a discontinuous control structure to guarantee perfect tracking for a class of systems satisfying “matching” conditions. Retaining the concept of an “attractive” surface but elimi- nating the control discontinuities, the method of sliding control [21] is currently being applied in many different applications. The works of Brockett [2], Hunt et al. [9], Isidori [10], Jakubczyk and Respondek [12] initiated a surge of interest in feedback lineariza- tion and more generally in the application of differential geometry to nonlinear control [10], [18], [20]. Recently, the area of robust nonlinear control has received a great deal of attention in the literature. Many methods employ a synthesis approach where the controlled variable is chosen to make the time derivative of a Lyapunov function candidate negative definite. Corless and Leitmann [3] have applied this approach to open-loop stable mis- matched nonlinear systems. A design methodology that has received a great deal of interest re- cently is “integrator backstepping.” The recent book by Krstic et al. [16] develops the backstepping approach to the point of a step by step design procedure. The integrator backstepping (IB) technique suffers from the problem of “explosion of terms.” The following example il- lustrates the backstepping approach as well as the difficulty that this paper seeks to solve: _ x I = f I ( x I )+ x P + f I ( x I ) _ x P = u (1) Manuscript received March 20, 1998; revised August 15, 1999. Recom- mended by Associate Editor, M. Krstic.
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