Stabilization of underactuated nonlinear systems to non‐feasible curves

This paper focuses on the development of stability conditions for systems of nonlinear non‐autonomous ordinary differential equations and their applications to control problems. We present a novel approach for the study of asymptotic stability properties for nonlinear non‐autonomous systems based on considering a parameterized family of sets. The proposed approach allows to state asymptotic stability conditions for a family of sets representing the level sets of a time‐varying Lyapunov function and to estimate the rate of convergence of solutions to a prescribed neighbourhood of the given curve. The obtained stability results are applied to the trajectory tracking problem for a class of nonholonomic systems.


Introduction and Problem Statement
Consider a system of nonlinear non-autonomous ordinary differential equationṡ where x ∈ R n , t ∈ R + = [0, ∞), f : R n × R + → R n . Assume that there exists a function x * ∈ C 1 (R + ; R n ) such that f (x * (t), t) = 0 for each t ∈ R + . Note that x * (t) is not a solution of (1) in general. In this paper, we propose novel asymptotic stability conditions for system (1) in a neighborhood of the curve x * (t). It should be noted that stability properties of such systems have been already studied in several publications, e.g., [5,6,8,9]. Unlike the above-mentioned results, we consider a general class of systems without assuming that ẋ * (t) → 0 as t → ∞. Our approach is based on considering a parameterized family of sets: let V t ⊂ R n , t ∈ R + , be a one-parameter family of non-empty sets. For a δ > 0, we denote the δ-neighborhood of the set V t at time t as B δ (V t )= y∈Vt {x∈R n : x−y <δ}, andB δ (V t ) is the closure of B δ (V t ). Definition 1.1 A family of sets V t is said to be locally uniformly asymptotically stable for (1) if: -it is uniformly stable, i.e. for each ε>0 there are δ>0 such that, for all t 0 ∈R + , if x 0 ∈B δ (V t0 ) then the corresponding solution of (1) satisfies x(t)∈B ε (V t ) for all t ≥ t 0 ; -δ-uniformly attractive with someδ>0, i.e. for each ε>0 there exists a t 1 ∈[0, ∞) such that, for all t 0 ∈R + , if x 0 ∈Bδ(V t0 ) then the corresponding solution of (1) satisfies x(t)∈B ε (V t ) for all t≥t 0 +t 1 ; If the attractivity property holds for everyδ>0, then the family of sets V t is called globally uniformly asymptotically stable for (1).
In this paper, we consider a family of sets representing the level sets of a time-varying Lyapunov function [2,7]. The obtained results extend the stability conditions obtained in [2,3] for gradient-like systems to general nonlinear non-autonomous systems (1).

Stability Conditions
The basic stability result of this paper is as follows.
Lemma 2.1 Let the following conditions be satisfied for system ) ≤ λ}, t ∈ R is locally uniformly asymptotically stable for system (1).

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Section 20: Dynamics and control The proof goes along the same line as the proof of [2,Theorem 3], and thus we omit it due to space limitations. The proof techniques rely on a careful analysis of the behavior of the time-varying function V (x − x * (t)) along the trajectories of system (1).
Note that the proposed result serves as a basis for obtaining various stability conditions, depending on properties of f , which we plan to present in future works. In the next section, we demonstrate an application of the obtained stability conditions to several control problems. In particular, we present a constructive solution to the trajectory tracking problem for a class of nonholonomic systems.

Stabilizing Control Strategies
Consider a class of driftless control systems of the forṁ The vector fields f i (x) are assumed to be linearly independent in a neighborhood of a given curve x * (t) and satisfying the condition , and with some sets of indices S k ⊆ {1, 2, ..., m} k (k = 1, 2, 3) such that |S 1 | + |S 2 | + |S 3 | = n. It is well-known that any equilibrium of (2) cannot be stabilized by a regular time-invariant feedback law [1]. To stabilize (2) in a neighborhood of x * (t), we define control functions in the following way: +δ k 3 cos 2πκ 1 1 2 3 t sin 2πκ 2 1 2 3 t , i = 1, 2, ..., m.