Feedback Linearization
Also known as: Exact Linearization, Nonlinear Feedback Control, Input-Output Linearization
Feedback Linearization is a nonlinear control technique that uses a nonlinear state-feedback transformation to convert a nonlinear system into a linear one, enabling the use of standard linear control methods. Developed by Isidori, Sontag, and others in the 1980s, feedback linearization is conceptually elegant and powerful: if the system satisfies certain structural conditions (relative degree, decoupling matrix rank), the nonlinearities can be exactly cancelled through feedback, reducing the problem to linear design.
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When to use it
Use feedback linearization when the system is nonlinear but satisfies the relative degree and decoupling conditions, and when you want to leverage the power of linear control design for a nonlinear system. It is ideal for systems with known, smooth nonlinearities (e.g., electrical machines, aeronautical vehicles). Avoid feedback linearization if the system has uncertain or unmodeled nonlinearities (will fail to linearize), if the relative degree is not well-defined (e.g., algebraic loops), or if the zero-dynamics are unstable (inverted pendulum on a cart is a classical example).
Strengths & limitations
- Exact cancellation of nonlinearities (for systems satisfying relative degree conditions); not an approximation.
- Transforms the design problem to linear control, enabling use of mature linear methods (pole placement, LQR, H∞).
- Provides clear conditions (relative degree, full rank) for when exact linearization is possible.
- Can handle multivariable nonlinear systems elegantly through MIMO input-output decoupling.
- Theoretical framework is well-established and supported by strong algebraic geometry foundations.
- Requires perfect knowledge of the system model; uncertain nonlinearities prevent exact linearization and degrade robustness.
- Zero-dynamics stability is not guaranteed; some systems (e.g., nonminimum-phase systems) have unstable internal dynamics.
- Only handles systems with a specific structure (relative degree, full decoupling matrix); not all nonlinear systems qualify.
- Can lead to unbounded control signals if the system approaches singular configurations or constraints.
- Requires state measurement or accurate estimation; feedback linearization is sensitive to measurement noise.
Frequently asked
What is relative degree and why is it important?
Relative degree is the number of times you must differentiate an output to make the input appear explicitly. For example, if y = h(x) and you need to differentiate r times before u appears in y^(r), the relative degree is r. Relative degree must equal the system order for exact feedback linearization to be possible. If relative degree < system order, you have internal (zero) dynamics that must be checked for stability.
What are zero-dynamics and why must they be stable?
Zero-dynamics are the internal dynamics that remain hidden when the input-output map is linearized. They represent the system behavior when the output is constrained to be zero. For the closed-loop system to be stable, zero-dynamics must be stable; if not, the linearized design may stabilize the output but destabilize internal states (e.g., energy building up in unobserved modes).
How does feedback linearization differ from linearization around an equilibrium?
Local linearization (Taylor expansion) only approximates the system near an equilibrium and is valid for small perturbations. Feedback linearization is global (on a large domain) and exact—it cancels nonlinearities through feedback. This is far more powerful, but it requires the system to satisfy strong structural conditions (relative degree, non-singular decoupling matrix).
What happens if the decoupling matrix is singular?
A singular decoupling matrix means the nonlinear system does not decouple into independent SISO (single-input single-output) loops via feedback. This indicates the system has coupling between inputs and outputs that cannot be overcome through feedback alone. You must either redesign the control structure or use alternative methods like sliding mode or MPC.
Sources
- Isidori, A. (1995). Nonlinear Control Systems (3rd ed.). Springer-Verlag. DOI: 10.1007/978-1-84628-615-5 ↗
- Sontag, E. D. (1983). A concept of input-output linearization. Proceedings of the 22nd IEEE Conference on Decision and Control. link ↗
- Nijmeijer, H., & Van der Schaft, A. J. (1990). Nonlinear Dynamical Control Systems. Springer-Verlag. DOI: 10.1007/978-1-4757-2101-0 ↗
How to cite this page
ScholarGate. (2026, June 3). Feedback Linearization. ScholarGate. https://scholargate.app/en/control-theory/feedback-linearization
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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- H-infinity ControlControl Theory↔ compare
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- Sliding Mode ControlControl Theory↔ compare