Nonsmooth feedback stabilizer for strict-feedback nonlinear systems that may not be linearizable at the origin
Introduction
In practice, there exist systems that do not have the first approximation at the origin, e.g., a leaky bucket whose dynamics is given by [13, p. 41] or the hydraulic control systems [15]. Partly motivated by this fact, we construct a continuous (but possibly nondifferentiable) state feedback stabilizer which globally stabilizes a single-input nonlinear system in the strict-feedback form given bywhere , , are functions vanishing at the origin and 's are rational numbers whose numerators and denominators are all positive odd integers (we will call such a positive odd rational number). We stress that if , then the system is not linearizable at each point with as well as at the origin, hence the standard backstepping design, which requires smoothness of the vector field, does not work directly.
This is a sharp contrast to the previous works [2], [3], [4], [9], [11], [14] which have considered a system whose right-hand side is continuously differentiable in the state , or all 's are greater than or equal to 1 so that its linearization at the origin may be uncontrollable. In [9], [11], [14], they propose a state feedback controller for the system (1) in which all 's are positive odd integers. Lin and Qian [9] explicitly construct, using a tool called adding a power integrator, a globally stabilizing smooth feedback control law for system (1) under the condition that the odd integer powers are in decreasing order (i.e., ), and under a growth condition that , , where each is a smooth nonnegative function. The decreasing assumption and the growth condition have been removed in [4], [11] while a continuous (instead of smooth) feedback is obtained in [11] and a smooth but time-varying feedback is designed in [4]. More generally, a triangular systemis studied in [2], [3], but it is assumed that all 's are so that its linearization at the origin does exist.
In this paper, we design a state feedback control law as well as a (positive definite and proper) Lyapunov function to make the origin globally asymptotically stable (GAS). The control law has an interesting feature: it contains some exponents (or powers) which are determined by a set of inequalities during the design procedure (thus, they are design parameters). These exponents are also closely related to the Lyapunov function used to prove the stability. The design procedure shows its efficiency when we discuss the finite time stabilization problem [1], [5] since a slight change of the inequalities involved in the asymptotic stabilization problem ensures the existence of finite time stabilizer. One drawback of the proposed existence condition (the inequalities) is that it is not easy to check in general (although it can be converted to linear matrix inequality (LMI)). Thus, in order to avoid further difficulties, we provide explicit design guidelines for some special cases of 's.
The paper is organized as follows. In Section 2.1, we state our main theorem for global stabilization whose proof is given in Section 2.2. Global finite time stabilization problem is discussed in Section 2.3. In Section 2.4, the conditions proposed in the main theorems of Sections 2.1 and 2.3 are discussed in detail, where some relation to the previous work [11] is also pointed out. Finally, we conclude the paper in Section 3.
For convenience, let us define the set of all rational numbers whose numerators and denominators are all positive odd integers by . Note that the set is closed under multiplication, division and odd number of additions, but is not closed under even number of additions or subtraction, and that, therefore, or for , , is a positive definite function of .
Section snippets
Statement of main theorem
We now state our main theorem. Theorem 1 Suppose that, for the system (1), , . If there exist such thatthen there exists a feedback controller with which renders the origin of the closed loop system GAS. In addition, if the assumption holds with all , then a smooth feedback controller exists. Remark 1 Note that from (4), the condition () implies
Conclusion
In this paper, we have developed a state feedback design method for the powers of the integrators perturbed by lower triangular vector fields. The result extends the work [11] in the sense that the powers are not restricted to positive odd integers. Although the system considered here may not have the first order approximation around the origin, which has rarely been studied in the literature, the proposed controller guarantees GAS of the origin. The finite time stabilization problem is
Acknowledgments
The authors would like to thank the reviewers for their constructive and pertinent comments, in particular, for suggesting the finite time stabilization problem which has not been discussed in the original version.
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