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Nonadaptive Learning in Robust Nonlinear Output Regulation
Shimin Wang, Martin Guay, Richard D. Braatz
TL;DR
Robust regulation of general nonlinear output-feedback systems with arbitrary relative degree remains challenging because nonlinearities make linear feedforward and internal-model designs inadequate. This paper develops a nonadaptive stabilization framework and establishes global asymptotic regulation, with convergence demonstrated for a controlled Duffing system.
Problem
General nonlinear output-feedback regulation with arbitrary relative degree remains challenging because nonlinear steady-state errors make linear feedforward and internal-model designs inadequate.
Method
The paper combines an input-driven filter with nonadaptive stabilization and recursive control to recast regulation as robust input-to-state stabilization.
Results
Global asymptotic stability is established under the stated assumptions, while a controlled Duffing example shows tracking and parameter-estimation errors converging to zero.
Takeaways & Limitations
The framework supports nonadaptive robust regulation for general nonlinear output-feedback systems with Input-to-State Stable internal dynamics.
Takeaways & Limitations
Existing generic internal-model methods depend on an unknown steady-state input mapping and often require numerical approximation with additional tuning.
Abstract
from arXiv · showhide
This paper considers robust nonadaptive regulation for general nonlinear systems in an output-feedback setting with arbitrarily high relative degree. We develop a nonadaptive design that combines an input-driven filter and a generic internal model with a recursive backstepping law, thereby recasting the regulation problem as the robust input-to-state stabilization of an augmented error system. Unlike adaptive schemes, the proposed method does not rely on linearly parameterized regressors and does not require the construction of Lyapunov functions having merely nonpositive derivatives. Under standard assumptions on the exosystem, including purely imaginary and simple eigenvalues, together with a minimum-phase input-to-state stability condition on the internal dynamics, we establish global asymptotic regulation and derive explicit, verifiable inequalities for selecting the design gains. The resulting nonadaptive framework guarantees convergence of the estimation and tracking errors even when the controlled-system dynamics are complex or only partially known. The effectiveness of the theoretical results is demonstrated using a benchmark controlled Duffing system.
1. Introduction.
The introduction motivates robust nonlinear output regulation and reviews internal-model approaches, emphasizing the limitations of adaptive and existing nonadaptive methods. It then positions the article’s nonadaptive learning framework for general nonlinear output-feedback systems with arbitrary relative degree.
- Motivation: Output regulation tracks desired signals while rejecting external disturbances, but nonlinear plants produce steady-state errors that depend nonlinearly on exogenous signals.This makes linear feedforward or linear regulation techniques insufficient for general nonlinear plants.
- Related work: Internal-model structures have been developed for uncertain linear exosystems and nonlinear exosystems generating non-sinusoidal signals.The introduction cites canonical linear internal models and nonlinear internal models as major approaches to nonlinear output regulation.
- Limitations of adaptive methods: Adaptive internal-model methods require Lyapunov functions with non-positive derivatives and explicit regressors, limiting robustness and applicability to suitable parametric uncertainty.Counterexamples show boundedness can fail even under small external inputs, while the required regression form restricts generality.
- Limitations of nonadaptive methods: Nonadaptive methods avoid explicit parameter adaptation, but generic internal models still require an unknown nonlinear mapping for the steady-state input.Existing solutions often approximate this mapping numerically using least-squares techniques, introducing additional tuning requirements.
- Contribution: The article addresses robust output regulation for general nonlinear output-feedback systems with arbitrary relative degree using nonadaptive learning methods.The arbitrary-relative-degree setting is more challenging than relative degree one because regulated-output derivatives are unavailable and internal-model and stabilization structures must be redesigned.
2. Problem Formulation and Assumptions.
The paper formulates robust output regulation for nonlinear systems with uncertain exosystem and plant parameters, then introduces assumptions and constructions supporting a generic internal model. The formulation requires bounded closed-loop solutions with vanishing tracking error under simple, purely imaginary exosystem eigenvalues and minimum-phase internal dynamics.
- 2. Problem Formulation and Assumptions.: The plant combines a fully nonlinear z-subsystem with a partially structured linear x-subsystem, while tracking error, input, reference output, and uncertain parameters are explicitly defined.The state is partitioned as (z, x), with x having relative degree r≥1 and nonlinear terms depending on z, y, v, and w.
- 2. Problem Formulation and Assumptions.: The regulation problem requires a control law that preserves global solution existence and boundedness while making the tracking error converge to zero for all admissible initial conditions and uncertainties.The stated problem quantifies over initial exosystem states, compact parameter sets containing the origin, and arbitrary plant initial states.
- 2. Problem Formulation and Assumptions.: The exosystem uncertainty assumption restricts S(σ) to have simple eigenvalues with zero real parts, covering constant and multi-tone sinusoidal reference signals.The associated exogenous signals may have unknown initial phases and amplitudes, while their frequencies are arbitrarily known.
- 2. Problem Formulation and Assumptions.: A globally defined smooth regulator-equation solution z(v,w,σ) describes the z-subsystem steady state, with z(0,w,σ)=0.The function z(v,w,σ) is identified as the steady state of the z-subsystem.
- 2. Problem Formulation and Assumptions.: The minimum-phase condition requires the translated inverse z-system to be input-to-state stable with respect to its state deviation and tracking error input.The paper further states that this condition guarantees input-to-state stability and supports a continuous Lyapunov-like function through the changing supply function technique.
- 2. Problem Formulation and Assumptions.: For relative degree r≥2, the design introduces an input-driven filter whose estimate error is transformed into dynamics involving a Hurwitz matrix A=Ac−λCc.The filter state ˆx estimates x, and the transformed error components are defined by ˜x_i=b^-1x_i−ˆx_i.
- 2.1. Generic internal model design.: Under polynomial steady-state input and nondegeneracy assumptions, the generic internal-model design constructs a steady-state generator and a nonlinear mapping satisfying the required differential equations.The polynomial representation yields a monomial vector with imaginary-axis dynamics, while the generalized Sylvester framework supplies the generator construction and mapping.
3. Main results.
The main results establish a recursive backstepping controller that solves the robust output regulation problem under Assumptions 2.2–2.10 and yields global asymptotic stability. A corollary also provides an adaptive estimate of the design gain while retaining boundedness and convergence properties.
- Recursive design: Recursive backstepping iteratively designs the control law to ensure convergence, robustness, and stability of the augmented error system.The construction uses error variables ǫ1 = e and recursively defined virtual controls αi.
- Theorem 3.1: Theorem 3.1 guarantees a continuous positive definite function Ur for which the closed-loop equilibrium at the origin is globally asymptotically stable.The stability conclusion holds for all µ ∈ V × W × S.
- Corollary 3.2: Corollary 3.2 shows that replacing k∗ with an estimate ˆk preserves a controller solving Problem 2.1 using the recursively defined functions α1, α2, and αi.The result applies under the same Assumptions 2.2–2.10 with a sufficiently large positive smooth function ρ(·).
- Corollary 3.2: The adaptive gain estimate ˆk remains uniformly bounded and converges to a finite limit, while the resulting design remains non-adaptive in its control-law framework.The updated law does not generate an unbounded high-gain.
4. Application to Duffing’s system.
The Duffing-system simulation applies the proposed controller to a system with an uncertain sinusoidal disturbance and specified frequency range. With explicitly selected gains and initial conditions, tracking and parameter-estimation errors converge nearly to zero within 50 seconds while the control signal becomes periodic.
- System setup: The controlled Duffing system uses coefficients c1 = 1.5, c2 = −2, and c3 = 0.5, with an external disturbance of unknown amplitude, frequency, and phase.The disturbance is d(t) = A cos(σt + ψ) and can be generated by an uncertain exosystem.
- System setup: The output map is h(v, w) = v1, the unknown frequency satisfies σ ∈ [0.1, 1], and the simulation uses σ = 0.5.The exosystem state is restricted to V = {v ∈ R2 : ∥v∥ ≤ 2.1}.
- Controller implementation: The design selects ρ(e) = 2 + 2e2, λ1 = 4, λ2 = 4, and m1 through m8 to make the relevant inequalities and matrices Hurwitz.The selected values are m1 = 1, m2 = 5.1503, m3 = 13.301, m4 = 22.2016, m5 = 25.7518, m6 = 21.6013, m7 = 12.8005, and m8 = 5.2001.
- Simulation results: Nearly zero tracking error and zero steady-state parameter-estimation error are achieved within 50 seconds, while the control signal converges to a periodic signal.The simulation starts from x(0) = col(1, 1), v(0) = col(1, 2), ˆx(0) = 02, η(0) = 08, and ˆk(0) = 0.
5. Conclusion.
The article proposes a nonadaptive nonlinear robust output-regulation framework for general nonlinear output-feedback systems and illustrates it with a controlled Duffing system.
- Conclusion: The article proposes a nonadaptive nonlinear robust output-regulation approach for general nonlinear output-feedback systems with error output.The framework addresses systems formulated in an output-feedback setting with error output.
- Conclusion: The proposed framework transforms robust output regulation into a robust nonadaptive stabilization method for systems with Input-to-State Stable dynamics.This reformulation is the central methodological contribution described in the conclusion.
- Conclusion: A numerical controlled Duffing-system example illustrates the approach and shows convergence of the parameter estimation error.The conclusion identifies the Duffing system as the numerical demonstration of the framework.