What Is Feedback Linearization?

Feedback linearization is a nonlinear control technique that transforms a nonlinear system into an equivalent linear system by applying a precisely computed control law and a change of coordinates, enabling the full suite of linear controller design methods to be applied to systems whose natural dynamics are inherently nonlinear. The central idea is to select a feedback signal that cancels the nonlinear terms in the plant's equations of motion, leaving a linearized inner loop to which a conventional linear outer loop can then be attached. The approach differs fundamentally from classical linearization, which approximates the system's behavior around a single operating point using a Taylor series truncation; feedback linearization is exact across the full range of motion, provided the plant model is accurate.

The theoretical foundation was developed primarily by Alberto Isidori and Arthur Krener in the 1970s and 1980s, using the geometric language of differential geometry and Lie algebras. Their work established the necessary and sufficient conditions under which exact feedback linearization is possible, and introduced the concept of the relative degree of a nonlinear system. The ScienceDirect overview of feedback linearization summarizes the mathematical framework and its relationship to the broader class of nonlinear control methods.

Input-Output Linearization

The most practically applied form of feedback linearization is input-output linearization, in which the control objective is to render the relationship between a selected output and a new virtual input exactly linear, rather than requiring the entire state-space to transform linearly. The relative degree of the system, defined as the number of times the output must be differentiated before the input appears explicitly, determines how many state variables are directly shaped by the linearizing feedback. Once the relative degree is established, the control law is constructed by computing the Lie derivatives of the output along the vector fields of the system and inverting the resulting expression. The remaining state variables, which do not appear in the input-output chain, evolve according to zero dynamics, and the stability of those zero dynamics must be verified separately. MIT OpenCourseWare's lecture notes from the course Dynamics of Nonlinear Systems provide a detailed exposition of input-output linearization and zero dynamics analysis.

State-Space Transformation and Exact Linearization

Full-state feedback linearization requires finding a coordinate transformation that converts the entire state-space representation into Brunovsky normal form, a cascaded chain of integrators. The conditions for this transformation to exist are that the system be affine in the control input and that the distribution spanned by the system's control vector fields satisfy an involutivity condition. When these conditions are met, the transformation is computed systematically and the resulting linear system can be controlled with pole placement, LQR, or any other linear design method. When the conditions are only partially satisfied, partial linearization is still possible and useful. The SpringerLink reference on feedback linearization covers both the full and partial cases and discusses implementation in the presence of modeling uncertainty.

Practical Considerations

Feedback linearization relies on an accurate model of the plant: the cancellation of nonlinear terms is exact only when the mathematical model matches the real system precisely. Parametric uncertainty and unmodeled dynamics degrade the cancellation, potentially reintroducing nonlinear behavior or destabilizing the closed loop. Robust extensions of feedback linearization, including sliding mode control and adaptive feedback linearization, address this limitation by combining the linearizing transformation with parameter estimation or sliding surfaces that maintain performance in the presence of bounded uncertainty. Computational cost is another practical concern, since the control law involves real-time evaluation of Lie derivatives that can be computationally intensive for high-dimensional systems.

Applications

Feedback linearization has applications in a wide range of disciplines, including:

• Robotic manipulator control for trajectory tracking
• Spacecraft attitude control systems
• Autonomous vehicle path following and steering control
• Chemical reactor temperature regulation with nonlinear kinetics
• Biomedical insulin delivery systems with nonlinear glucose dynamics