Source-linked AI summary
Homogeneous Approximation, Recursive Observer Design, and Output Feedback
Vincent Andrieu, Laurent Praly, Alessandro Astolfi
TL;DR
The paper addresses global asymptotic output-feedback stabilization when nonlinear dynamics have distinct behavior near the origin and at infinity. It introduces bi-limit homogeneity and a recursive observer design, then combines them to obtain a new output-feedback stabilization result. The paper also develops associated stability and robustness results.
Problem
The paper seeks global asymptotic output-feedback stabilization for systems whose nonlinear term has distinct nonzero behavior at the origin and infinity.
Method
It combines homogeneity in the bi-limit, a recursive observer for integrator chains, and homogeneous-in-the-bi-limit state-feedback design.
Results
The combination yields a new global asymptotic stabilization result by output feedback for systems whose dominant part is a chain of integrators.
Takeaways & Limitations
The two tools provide a framework for nonlinear observer and output-feedback design across both the origin and infinity regimes.
Takeaways & Limitations
In general, homogeneity in the limit does not imply corresponding homogeneity of the derivative.
Abstract
from arXiv · showhide
We introduce two new tools that can be useful in nonlinear observer and output feedback design. The first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity (homogeneity in the bi-limit). Exploiting this extension, we give several results concerning stability and robustness for a homogeneous in the bi-limit vector field. The second tool is a new recursive observer design procedure for a chain of integrator. Combining these two tools, we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems.
1. Introduction.
The paper addresses global asymptotic stabilization by output feedback for nonlinear systems whose dynamics include both linear behavior near the origin and polynomial growth at infinity. It introduces homogeneity in the bi-limit and a recursive observer to handle these two regimes together.
- Motivation: The target is a globally stabilizing output-feedback controller for the nonlinear system with measured output y = x1.The illustrative dynamics contain an integrator chain, control input u, and nonlinear term δ2.
- Related approaches: Existing domination-based designs treat δ2 as a perturbation and design the controller for the corresponding linear system.Such designs rely on robustness of closed-loop global asymptotic stability to the nonlinear disturbance.
- Related approaches: Earlier results cover q = 1 with c∞ = 0 using linear output feedback and p ≥ 1 with c0 = 0 using homogeneous output feedback.These cases do not simultaneously include nonzero coefficients at the origin and infinity.
- Contributions: The paper develops bi-limit homogeneity, stability and robustness results, a recursive observer for integrator chains, and a compatible state-feedback design.These components are combined for global asymptotic output-feedback stabilization.
- Problem structure: When |x2| is small, δ2 is approximated by c0 x2, while for large |x2| it is approximated by c∞x2^p.The resulting combination of linear and polynomial terms motivates a two-regime homogeneity framework.
2. Homogeneous approximation.
This section extends homogeneous approximation to both the origin and infinity, allowing different weights and degrees in the two limits. The resulting framework supports stability, robustness, ISS, small-gain, and finite-time convergence results.
- Examples: For the illustrative nonlinearity, the 0-limit approximation is c0 x2^q and the ∞-limit approximation is c∞x2^p.The associated vector field is homogeneous in the bi-limit when 0 < q < p < 2.
- Definitions: Homogeneity in the bi-limit means that a function or vector field is homogeneous both in the 0-limit and in the ∞-limit.The two approximations may use distinct weights, degrees, and approximating functions.
- Robustness: The stability properties remain under perturbations that do not change the relevant approximating homogeneous function.At infinity, boundedness is preserved for perturbations negligible relative to the dominant homogeneous part.
- Stability: The framework provides Lyapunov functions for vector fields whose homogeneous approximations at both limits have globally asymptotically stable origins.The resulting function is C1, positive definite, and proper.
- Robustness: The same assumptions yield an input-to-state stability property with respect to disturbances.The paper also establishes a small-gain result for the resulting systems.
- Finite-time convergence: If d∞ > 0 > d0, all solutions converge to the origin in finite time uniformly in the initial condition.This extends finite-time convergence results based on a globally asymptotically stable origin in the 0-limit approximation.
3. Recursive observer design for a chain of integrators.
The paper introduces a recursive observer design for integrator chains using homogeneous-in-the-bi-limit error injections. The construction yields global observation for the chain and for its approximations at the origin and infinity.
- Setup: The observer is designed for the chain ˙Xn = SnXn + Bnu with measured state component X1.The design proceeds recursively from the last state toward the first.
- Observer construction: The proposed observer is exact for a chain of integrators with any input u.This distinguishes it from the cited observer designs discussed in the paper.
- Observer construction: The output injection K1 is selected as a homogeneous-in-the-bi-limit vector field whose error system and both approximations are globally asymptotically stable.The error dynamics are ˙E1 = SnE1 + K1(e1), together with corresponding 0-limit and ∞-limit systems.
- Recursive step: Theorem 3.1 extends a stable lower-dimensional error system to a higher-dimensional one through a recursive construction.A sufficiently large parameter ℓ makes the Lyapunov derivative negative definite for the system and its homogeneous approximations.
- Result: Iterating the construction produces a homogeneous-in-the-bi-limit observer that globally observes the chain and its origin and infinity approximations.The result is stated through global asymptotic stability of the three error systems in (3.12).
- Example: For the two-dimensional chain, the observer and both homogeneous approximations are global observers of the same system.The example uses positive parameters ℓ1 and ℓ2 in the recursive injection.
4. Recursive design of a homogeneous in the bi-limit state feedback.
This section extends homogeneous state-feedback backstepping to homogeneity in the bi-limit for chains of integrators. The recursive construction preserves global asymptotic stability for the system and its two homogeneous approximations.
- State-feedback extension: The paper constructs a homogeneous-in-the-bi-limit feedback φn for the full integrator chain.The closed-loop system has specified weights and degrees at the origin and at infinity.
- Recursive backstepping: The design extends a stabilizing feedback from an i-dimensional integrator chain to dimension i + 1.The auxiliary systems use shift matrices Si and input vectors Bi.
- Recursive backstepping: Theorem 4.1 assumes a homogeneous-in-the-bi-limit stabilizer with suitable differentiability and stable approximating systems.It then provides a higher-dimensional feedback with corresponding approximating functions.
- Stability proof: The proof combines a homogeneous-in-the-bi-limit Lyapunov function with a gain parameter whose sufficiently large value makes its derivative negative definite.The same argument is applied to the system and its homogeneous approximations.
1. Construction of the Lyapunov function.
The construction recursively builds a positive definite, proper, C1 Lyapunov function whose partial derivatives are homogeneous in the bi-limit. The design uses degree conditions and a sign-preserving auxiliary function to extend the construction along the integrator chain.
- dV0 and dV∞ are chosen larger than the maximum origin and infinity weights, respectively, enabling construction of a C1, positive definite, proper Lyapunov function.
- Each partial derivative of Vi is homogeneous in the bi-limit, with degrees dV0−r0,j and dV∞−r∞,j for the origin and infinity approximations.
- The recursive Lyapunov function Vi+1 is defined on the extended integrator state and remains positive definite and proper under the degree conditions.
- The function Vi+1 is C1 because ψi(Xi)=φi(Xi)^αi is C1, and its partial derivatives retain homogeneity in the bi-limit.
- The auxiliary function ψi+1 is chosen C1, homogeneous in the bi-limit, and sign-consistent with Xi+1−φi(Xi), supporting the recursive control construction.
3. Selection of k.
The recursive design selects the gain k sufficiently large to establish global asymptotic stability, while preserving robustness under sufficiently small perturbations. Special parameter choices recover simpler designs and a previously known construction.
- k is selected above a threshold k* so the relevant inequalities hold for every nonzero extended state, implying global asymptotic stability.
- At the end of the recursion, the origin is a globally asymptotically stable equilibrium for the resulting systems.
- When d0 and d∞ are nonnegative, setting αi=1 is allowed; under additional degree orderings, simpler choices of αi and ψi are available.
- When 0≤d0≤d∞, the design with αi=1 recovers the construction in.
- The construction yields a positive gain k2 for the two-dimensional chain-of-integrators example, given any k1>0.
- For sufficiently small perturbation coefficients |c0| and |c∞|, a positive bound cG preserves global asymptotic stabilization by the designed control law.
5. Application to nonlinear output feedback design.
The paper combines homogeneity in the bi-limit with recursive observer and state-feedback designs to obtain global asymptotic output-feedback stabilization for feedback- and feedforward-form systems whose dominant dynamics are a chain of integrators.
- Motivation and design framework: The approach targets nonlinear systems whose dominant dynamics can be represented by a chain of integrators, extending a long-standing domination-based output-feedback strategy.Earlier approaches treated nonlinear terms as perturbations of a linear design; the proposed method instead exploits homogeneity in the bi-limit together with recursive designs.
- Motivation and design framework: The proposed output-feedback construction uses a homogeneous in the bi-limit state feedback φn, a homogeneous in the bi-limit observer vector field K1, and a positive gain L.The recursive procedures generate continuous functions φn and K1, with the observer state estimate ˆXn in R^n.
- Stability result: The resulting closed-loop origin is globally asymptotically stable for the system and its homogeneous approximations.The proof constructs the feedback and observer through systems whose origins are globally asymptotically stable, then applies robustness results for homogeneous approximations.
- Feedback-form stabilization: For feedback form, selecting d0 ≤ d∞ yields class-KL bounds on both the state x and observer state ˆXn when L is sufficiently large.The bounds hold for trajectories satisfying the plant, observer, and disturbance relations, with disturbance influence represented through class-K and class-KL functions.
- Feedforward-form stabilization: For feedforward form, selecting d∞ ≤ d0 yields the same class-KL bounds when L is sufficiently small.The corresponding gain range is L in (0,L*], and the bounds apply to both x and ˆXn.
- Scope and examples: The combined tools provide global asymptotic output-feedback stabilization for both feedback and feedforward forms, including systems with nonzero behavior at the origin and infinity.The feedback-form result uses large L under -1 < d0 ≤ d∞ < 1, while the feedforward-form result uses small L under -1 < d∞ ≤ d0 < 1.
6. Conclusion.
The paper develops homogeneous-in-the-bi-limit Lyapunov and approximation tools to establish stability and robustness, and applies them to output-feedback observer design. The resulting conditions support global asymptotic stabilization for the considered systems.
- Continuity and homogeneity properties are used to control approximating functions near zero and on compact sets away from the origin.The proof repeatedly exploits compactness, continuity, and homogeneity in the zero-limit and infinity-limit cases.
- The resulting stability analysis yields global asymptotic stability of an invariant compact set for the system at infinity.The compact set is identified through the infinity-limit Lyapunov construction.
- The proof combines Lyapunov functions for the origin limit, infinity limit, and the global system into one proper C1 Lyapunov function.Its derivative is then shown to be negative definite away from the origin.
- Homogeneous-in-the-bi-limit arguments establish negativity conditions for both the system and its homogeneous approximations.The zero- and infinity-limit inclusions provide the basis for applying the stability lemmas.
- The robustness analysis derives a linear gain property and then uses a small-gain argument to obtain the state bound.The construction produces a class-KL bound for the state from the available inequalities.