Strict-feedback form

Abstract: A method is proposed for designing controllers with arbitrarily small tracking error for uncertain, mismatched nonlinear systems in the strict feedback form. This method is another "synthetic input technique," similar to backstepping and multiple surface control methods, but with an important addition, /spl tau/-1 low pass filters are included in the design where /spl tau/ 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 backstepping 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 boundedness of tracking error semiglobally, when the nonlinearities in the system are non-Lipschitz.

Abstract: A systematic procedure for the design of adaptive regulation and tracking schemes for a class of feedback linearizable nonlinear systems is developed. The coordinate-free geometric conditions, which characterize this class of systems, do not constrain the growth of the nonlinearities. Instead, they require that the nonlinear system be transformable into the so-called parametric-pure feedback form. When this form is strict, the proposed scheme guarantees global regulation and tracking properties, and substantially enlarges the class of nonlinear systems with unknown parameters for which global stabilization can be achieved. The main results use simple analytical tools, familiar to most control engineers. >

Abstract: The dynamic surface control (DSC) technique was developed recently by Swaroop et al. This technique simplified the backstepping design for the control of nonlinear systems in strict-feedback form by overcoming the problem of "explosion of complexity." It was later extended to adaptive backstepping design for nonlinear systems with linearly parameterized uncertainty. In this paper, by incorporating this design technique into a neural network based adaptive control design framework, we have developed a backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. Our development is able to eliminate the problem of "explosion of complexity" inherent in the existing method. In addition, a stability analysis is given which shows that our control law can guarantee the uniformly ultimate boundedness of the solution of the closed-loop system, and make the tracking error arbitrarily small.

Abstract: Implementation of adaptive backstepping controllers requires analytic calculation of the partial derivatives of certain stabilizing functions. It is well documented that, as the order of a nonlinear system increases, analytic calculation of these derivatives becomes prohibitive. Therefore, in practice, either alternative control approaches are used or the derivatives are neglected in the implementation. Neglecting the derivatives results in the loss of all guarantees proven by Lyapunov methods for the adaptive backstepping approach and may result in instability. This paper presents a new implementation approach for adaptive backstepping control. The main objectives are to facilitate the derivation and implementation of the adaptive backstepping approach, with performance guarantees proven by Lyapunov methods, for applications that were prohibitively difficult using the standard analytic implementation approach. The new approach uses filtering methods to produce certain command signals and their derivatives which eliminates the requirement of analytic differentiation. The approach also introduces filters to generate certain compensating signals necessary to compute compensated tracking errors suitable for adaptive parameter estimation. We present a set of Lemmas and Theorems to analyze the performance both during the initialization and the operating phases. We show that the initialization phase is of finite duration that can be controlled by selection of a design parameter. We also show that all signals within the system are bounded during this short initialization phase. During the operating phase, we show that the command filtered implementation approach has theoretical properties identical to those of the conventional approach. The general approach is presented and analyzed for systems in generalized parameter strict feedback form. Extensions of the approach are presented to demonstrate the application of the method to a land vehicle trajectory following application. Application and effectiveness of the proposed method is shown by simulation results.