panteley1998global

On Global Uniform Asymptotic Stability of Nonlinear Time-varying Systems in Cascade

Authors: Panteley, Loria · Year: 1998 · Venue: Systems and Control Letters
Raw: md

Summary

This paper gives Lyapunov-based sufficient conditions under which the cascade of two nonlinear time-varying (non-autonomous) systems preserves global uniform stability (GUS) or global uniform asymptotic stability (GUAS). The driving system perturbs the driven system through an interconnection term g(t,x)x_2; the central tool is an absolute-integrability condition on the perturbing trajectory x_2(t) together with growth restrictions on the Lyapunov function of the driven subsystem. It extends the authors’ earlier ECC’97 result, which required the perturbing system to be globally exponentially stable (GES), to the weaker GUAS + integrability setting.

Key Claims

• Theorem 1 (GUS preserved): If the unforced driven system ẋ₁ = f₁(t,x₁) is GUS with a Lyapunov function V(t,x₁) satisfying the growth bounds (7)–(8), the interconnection term satisfies the at-most-linear-in-x₁ growth bound (9), and the driving system ẋ₂ = f₂(t,x₂) is GUAS with ∫_{t₀}^∞ ‖x₂(t)‖ dt ≤ φ(‖x₂(t₀)‖) for some φ ∈ K (10), then the cascade (4)–(5) is GUS.
• Theorem 2 (GUAS preserved): Under the same Assumptions A2–A3, if in addition ẋ₁ = f₁(t,x₁) is GUAS (with V satisfying (7)), then the full cascade is GUAS.
• Theorem 3 (integrability relaxed): If the driving system (5) is GUAS and the driven system admits a Lyapunov function with the stiffer K∞ sandwich and growth conditions A4 [(12)–(16)] plus the interconnection bound A5 [(17): ‖g(t,x)‖ ≤ θ(‖x₂‖)‖x₁‖, θ continuous nondecreasing], then the cascade is GUAS — without assuming x₂ ∈ L₁.
• Assumption A2 imposes no restriction on how x₂ enters, but requires g(t,x) to grow at most linearly in x₁.
• The integrability condition (10) can be weakened when g(t,x) = φ₁(t,x)φ₂(t,x₂) factorizes: it then suffices that φ₂(t,x₂)x₂ ∈ L₁ (11), which for the worked example reduces to the milder requirement x₂ ∈ L₂.
• GUAS of the cascade follows as a corollary of Theorem 2 when Σ₂ is GES (recovering the earlier [13] result).

Method

The systems treated are the non-autonomous cascade

${\Sigma }_1:{\dot{x}}_1=f_1(t,x_1)+g(t,x)x_2,{\Sigma }_2:{\dot{x}}_2=f_2(t,x_2)$

with x₁ ∈ ℝⁿ, x₂ ∈ ℝᵐ, x ≜ col[x₁,x₂]. Proofs use the second method of Lyapunov on V(t,x₁) for the unforced driven subsystem ẋ₁ = f₁(t,x₁) (6). The cross term contributes

${\dot{V}}_{(4)}(t,x)=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_1}\cdot f_1(t,x_1)+\frac{\partial V}{\partial x_1}\cdot g(t,x)x_2,$

whose first two terms are ≤ 0 by GUS of (6), leaving a bound V̇ ≤ ‖∂V/∂x₁‖ ‖g‖ ‖x₂‖. The growth bound (7), ‖∂V/∂x₁‖‖x₁‖ ≤ c₁V, converts this into V̇ ≤ (c₁c₃V + c₄)‖x₂‖; separating variables and integrating gives a Grönwall/comparison bound (21) that stays finite precisely because ∫‖x₂‖dt ≤ λ < ∞ (10). This rules out finite escape and yields uniform boundedness; an ε–δ contradiction argument then gives uniform stability. Theorem 3 replaces integrability with the structural conditions (15)–(16) on the K∞ functions, using a KL bound (39) on x₂(t) and a time T after which θ(‖x₂‖)‖x₂‖ < k < k⋆ so that V̇ ≤ -(μ/k⋆)α₃(‖x₁‖) (41).

This is a pure systems-theory paper — no robot model, no manipulator. The free-flying vs free-floating distinction does not arise here; the contribution is a general stability tool. The relevant regime mapping is downstream: a free-flying manipulator’s tracking/observer architecture can often be cast as such a cascade.

Relevance to thesis

Cascade (or “cascaded-systems”) arguments are the standard rigorous route to proving GUAS of nonlinear time-varying tracking and observer-based controllers — exactly the structure that arises for a free-flying space manipulator whose closed loop decomposes into a controller error subsystem perturbed by a parameter/velocity-estimation error subsystem. Theorem 2 is the workhorse: prove each piece GUAS and check the integrability/growth conditions, rather than searching for a single monolithic Lyapunov function. This is the kind of pencil-and-paper-verifiable result a control-theorist examiner expects to see cited when claiming global (not merely local) uniform asymptotic stability of a time-varying closed loop, and it is the natural foundation before layering risk-aware analysis on top of a proven-stable nominal loop.

Connections

Topics: cascaded_systems · lyapunov_stability · trajectory_tracking

Key Equations / Quotes

“a globally uniformly stable (GUS) nonlinear time-varying (NLTV) system remains GUS when it is perturbed by the output of a globally uniformly asymptotically stable (GUAS) NLTV system, under the assumption that the perturbing signal is absolutely integrable” (Abstract)

Assumption A1 growth bound (7):
$\Vert \frac{\partial V}{\partial x_1}\Vert \Vert x_1\Vert \le c_{1}V(t,x_1)\forall \Vert x_1\Vert \ge \eta$

Assumption A2 interconnection bound (9):
$\Vert g(t,x)\Vert \le {\theta }_1(\Vert x_2\Vert )+{\theta }_2(\Vert x_2\Vert )\Vert x_1\Vert$

Assumption A3 integrability (10):
${\int }_{t_0}^{\infty }\Vert x_2(t,t_0,x_2(t_0))\Vert dt\le \varphi (\Vert x_2(t_0)\Vert ),\varphi \in \mathcal{K}$

Theorem 3 key growth conditions (15)–(16):
${\alpha }_4(\Vert x_1\Vert )\Vert x_1\Vert \le b_1{\alpha }_1(\Vert x_1\Vert )\forall \Vert x_1\Vert \ge \eta ,k_{\star }{\alpha }_4(\Vert x_1\Vert )\Vert x_1\Vert \le {\alpha }_3(\Vert x_1\Vert )\forall x_1\in {\mathbb{R}}^n$

resulting decay (41):
${\dot{V}}_{(4)}(t,x)\le -\frac{\mu }{k_{\star }}{\alpha }_3(\Vert x_1\Vert )$

Open Questions

• The paper notes (after Eq. 11) that the integrability condition “can be weakened in some cases, depending on the cascaded system in question.” A general characterization of which factorizations g = φ₁φ₂ admit the milder L₂-type condition is left open.
• Theorems give sufficient conditions only; necessity/tightness of the linear-growth restriction (A2) and the K∞ growth conditions (A4) is not addressed.
• No constructive guidance on the design problem (choosing the feedback) for the time-varying case — the paper is explicitly an analysis result.