<p>This paper develops a differential-geometric and Lyapunov-based framework for the stabilization of single-input underactuated Euler–Lagrange systems in the presence of both matched and mismatched disturbances. Starting from the control-affine representation, the normal form is constructed via the Pfaffian annihilator of the distribution <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$D=\text{span}\{g,[f,g]\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>=</mo> <mtext>span</mtext> <mo stretchy="false">{</mo> <mi>g</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">]</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, which is shown to be involutive for all mechanical systems with a single control input. This property ensures the existence of globally consistent internal coordinates in which the zero dynamics retain a Hamiltonian and symplectic structure. A structure-preserving feedback law is then derived that introduces damping through a projected energy term <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N\nabla H(\eta ),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mi mathvariant="normal">∇</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> thereby maintaining the mechanical character of the system while guaranteeing Lyapunov stability. Matched disturbances are shown to yield input-to-state stability, while mismatched perturbations lead to uniform ultimate boundedness. The proposed framework unifies geometric control, Hamiltonian preservation, and robustness analysis under a single Lyapunov energy inequality. Its effectiveness is demonstrated on the disturbed cart–pendulum benchmark, confirming both asymptotic and bounded-stability regimes. The approach provides a bridge between nonlinear geometric control and practical energy-based design for underactuated systems.</p>

Differential-geometric normal forms and structure-preserving control of single-input underactuated Euler–Lagrange systems under disturbances

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2026年2月17日

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Faical Mnif

摘要

This paper develops a differential-geometric and Lyapunov-based framework for the stabilization of single-input underactuated Euler–Lagrange systems in the presence of both matched and mismatched disturbances. Starting from the control-affine representation, the normal form is constructed via the Pfaffian annihilator of the distribution $D=\text{span}\{g,[f,g]\}$ $D = span {g, [f, g]}$ , which is shown to be involutive for all mechanical systems with a single control input. This property ensures the existence of globally consistent internal coordinates in which the zero dynamics retain a Hamiltonian and symplectic structure. A structure-preserving feedback law is then derived that introduces damping through a projected energy term $N\nabla H(\eta ),$ $N \nabla H (η),$ thereby maintaining the mechanical character of the system while guaranteeing Lyapunov stability. Matched disturbances are shown to yield input-to-state stability, while mismatched perturbations lead to uniform ultimate boundedness. The proposed framework unifies geometric control, Hamiltonian preservation, and robustness analysis under a single Lyapunov energy inequality. Its effectiveness is demonstrated on the disturbed cart–pendulum benchmark, confirming both asymptotic and bounded-stability regimes. The approach provides a bridge between nonlinear geometric control and practical energy-based design for underactuated systems.