EBIF: Exact Bilinearization Iterative Form for Control-Affine Nonlinear Systems

In this paper, we develop a novel framework, Exact Bilinearization Iterative Form (EBIF), for transforming a nonlinear control-affine system into an exact finite-dimensional bilinear representation. In contrast to most existing approaches which generally lead to an infinite-dimensional representation, the proposed EBIF approach yields an iterative procedure for constructing a finite set of smooth coordinate functions that define an embedding, enabling an exact bilinear representation of the original nonlinear dynamics. Leveraging tools from algebra and differential geometry, we establish both necessary and sufficient conditions for a nonlinear system to be exactly bilinearizable. We further illustrate how the EBIF-induced bilinear systems facilitate reachability analysis and control design. Through theoretical analysis and numerical simulations, we demonstrate the effectiveness of the EBIF framework and highlight its potential in simplifying control synthesis for nonlinear systems.

💡 Research Summary

The paper introduces a novel framework called Exact Bilinearization Iterative Form (EBIF) for converting control‑affine nonlinear systems into exact finite‑dimensional bilinear representations. Traditional approaches such as Carleman linearization or Koopman operator methods typically lift a finite‑dimensional nonlinear system to an infinite‑dimensional linear or bilinear system and then truncate, resulting in approximation errors and rapid growth of the state dimension. In contrast, EBIF provides a systematic, constructive procedure that yields a finite‑dimensional bilinear model that is mathematically equivalent to the original dynamics, with no approximation.

The authors start by defining “exact bilinearization”: a smooth embedding Ψ: ℝⁿ → ℝʳ (n < r < ∞) that maps the original state x to a lifted state z = Ψ(x) such that the transformed dynamics take the form ˙z = Az + ∑_{i=1}^m u_i B_i z, where A and B_i are constant matrices. They formalize the notion of Ψ‑related vector fields and prove that the drift and input vector fields of the original system must be push‑forwards of the linear vector fields in the lifted coordinates.

The core of EBIF lies in algebraic and geometric constructions based on Lie derivatives and module theory. The space of smooth functions on ℝⁿ is treated as a free module over the ring C^∞(ℝⁿ). For a given set of vector fields {f, g₁,…,g_m}, the Lie derivative L_τ of a module Γ is defined, and a module is said to be τ‑invariant if L_τ Γ ⊆ Γ. The authors show that if one can construct a finitely generated, jointly invariant module for all system vector fields, then a finite set of smooth functions can be generated iteratively by repeatedly applying Lie derivatives. This iterative process—EBIF—adds new functions to the generating set until the module stabilizes, which is guaranteed by the Noetherian property of the function ring.

The resulting stable generating set provides the components of the embedding Ψ. In the lifted coordinates, each component ψ_j satisfies ˙ψ_j = L_f ψ_j + ∑ u_i L_{g_i} ψ_j, and because the module is invariant, these Lie derivatives are linear combinations of the ψ_k themselves. Consequently, the lifted dynamics are exactly bilinear with constant matrices A and B_i.

The paper then leverages the exact bilinear model for reachability analysis and optimal control. Since bilinear systems admit tractable reachability computations using linear‑algebraic tools, the reachable set of the original nonlinear system can be obtained directly from the lifted bilinear system. Moreover, the Hamilton‑Jacobi‑Bellman equation for optimal control reduces to a bilinear form, dramatically simplifying numerical solution and enabling global optimal policies.

Simulation studies illustrate the theory. A unicycle robot model is exactly bilinearized by augmenting the state with cos θ, sin θ, and a constant, yielding a 6‑dimensional bilinear system. A nonlinear electrical circuit and a multi‑input multi‑output mechanical system are also treated, showing that EBIF produces low‑dimensional bilinear models (often with only modest increase in dimension) while preserving exact dynamics. Comparisons with Carleman truncations demonstrate that EBIF avoids truncation error and state‑explosion, achieving superior accuracy and computational efficiency.

The authors acknowledge limitations: the construction depends on an initial choice of generating functions, and for highly nonlinear or high‑order polynomial systems the invariant module may become large, leading to higher‑dimensional embeddings. They suggest future work on automated function selection, integration with data‑driven observable learning, and extensions to systems with delays or switching dynamics.

In summary, EBIF offers a rigorous, constructive pathway from a broad class of control‑affine nonlinear systems to exact finite‑dimensional bilinear models, opening new possibilities for analysis, verification, and control synthesis that were previously limited to approximate or infinite‑dimensional frameworks.