Non-smooth feedback stabilizer for strict-feedback nonlinear systems not even linearizable at the origin

Abstract

We present a continuous feedback stabilizer for nonlinear systems in the strict-feedback form, whose chained integrator part has the power of positive odd rational numbers. Since the power is not restricted to be larger than or equal to one, the linearization of the system at the origin may fail. Nevertheless, we will show that the closed-loop system is globally strongly stable with the proposed continuous (but, possibly not differentiable) feedback. We formulate a condition that enables our design by characterizing the powers of the given system. The condition also shows that our result is an extension of (Qian and Lin, Systems and Control Letters, 2001) where the power of odd positive integers has been considered.

A universal method for robust stabilization of nonlinear systems: unification and extension of smooth and non-smooth approaches

“…This ability to create a C 1 controller is in sharp contrast to the prior state feedback stabilization results of [2], [14], which could only guarantee a non-smooth controller in all cases (21), (22). Remark 4.2: Note that since Scenario 2 is a special case of Theorem 2.1, we can use the methodology of the first scenario as to not limit ourselves to a non-smooth solution when we are not restricted to one by the previously mentioned constraints.…”

“…which was previously only stabilizable by a non-smooth controller [2]. But by Theorem 2.1 and Assumption 2.1, system (25) can be globally stabilized by selecting τ 1 = 2, τ 2 = 2/3, τ 3 = 0 and r 1 = r 2 = 1, r 3 = 5, with σ = 5.…”

“…Remark 3.1: If we use Scenario 2 with each p i ∈ IR + odd , then the problem statement of [2], which also yields a nonsmooth solution, is additionally covered as another special case of our more general Theorem 2.1, thereby lifting the non-smooth stabilizer limitation for all related systems.…”

“…Secondly, and more significantly, this proposed methodology will allow for the possibility to obtain a C 1 controller, by static state feedback, for some systems which could previously only be stabilized by a C 0 controller. As an example, for the systeṁ x 1 = x 3 2 + x 2 1 ,ẋ 2 = u, the methods of [14], [2] can only produce a C 0 controller (for reasons discussed in [14]). Note that the smooth result of [11] is inapplicable since x 2 1 is not of the same order as x 3 2 .…”

“…The power order restriction and growth condition were lifted (with p i ≥ 1 odd integers) in [14], though the trade-off for such flexibility was non-smooth feedback stabilization. The technique from [14] was recently extended in [2] to allow for fractional odd powers less than one, with a non-smooth controller still utilized.…”