On Robust Approximate Feedback Linearization: Control with Two Gain-scaling Factors
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
On Robust Approximate Feedback Linearization: Control with Two Gain-scaling Factors Sang-Young Oh and Ho-Lim Choi* Abstract: In this paper, we consider a problem of global stabilization for a class of approximately feedback linearized nonlinear systems. In order to handle more nonlinearity over the existing results, we provide a new feedback controller with two gain-scaling factors and we show that more nonlinearity can be treated by our control scheme. Moreover, we analytically show that the considered nonlinear systems can be stabilized by the proposed controller. Through comparison practical examples, we illustrate the improved features of our proposed control method. Keywords: Approximate feedback linearization, gain-scaling factors, nonlinear systems, robust stabilization.
1.
INTRODUCTION
Up-to-date, control problems on approximately feedback linearized systems have been actively studied [1–13]. In the existing results, the authors often assume certain conditions on the perturbed nonlinearity. In [1–4], the authors assumed the lower triangular condition, whereas in [1, 5–7], they assumed the upper triangular(feedforward) condition. These structural restrictions regarding the nonlinearity are relaxed in [8–11]. In [8], a robust adaptive control method was proposed based on backstepping techniques for nonlinear systems with non-triangular nonlinearity. Their considered non-triangular nonlinearities include all states and the nonlinearities are bounded by known functions. In [9], a robust controller was proposed for nonlinear systems where their method allows either upper or lower triangular or non-triangular nonlinearity in a unified way. However, their proposed method may lead to rather conservative restriction in terms of controller design and allowable nonlinearity. In [10], a decentralized event-based triggering mechanism for a class of nonlinear systems was studied, but the local Lipschitz conditions have to be satisfied. In [11], a robust feedback linearization technique for nonlinear systems with unmodeled nonlinearity is studied. However, in order to achieve the asymptotic stabilization, it is required that the growth rate of the unmodeled nonlinearity to be small. In this paper, we further extend the control scheme by suggesting a new controller with two gain-scaling factors in order to treat more nonlinearity over [1, 4, 7, 9, 14–16]. We show that with our new control method, more gen-
eralized results can be achieved over [1, 4, 7, 9, 14–16]. Moreover, we show that the convergence rate of the states can increase by adjusting two-gain-scaling factors α and ε. Careful comparisons in terms of both analysis and practical examples are provided in order to illustrate the clear advantage of our control method. 2.
SYSTEM FORMULATION
We consider a class of nonlinear systems given by x˙1 = x2 + δ1 (t, x, u), .. . x˙n−1 = xn + δn−1 (t, x, u), x˙n = u + δn (t, x, u),
(1)
or in a matrix form as x˙ = Ax + Bu + δ (t, x, u),
(2)
where x = [x1 , ·
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