Feedback Linearization
Feedback linearization cancels a system's nonlinearities with a change of coordinates and control, exposing linear dynamics for standard design.
Turning nonlinear into linear
Feedback linearization transforms a nonlinear system into an equivalent linear one, exactly and not by approximation, through a nonlinear change of coordinates and a nonlinear feedback that cancels the plant's nonlinear terms. Once linear, the system is controlled with familiar linear methods, and the design is mapped back through the coordinate change. It differs fundamentally from linearizing about an operating point, which is only locally valid.
Relative degree and zero dynamics
The key structural quantity is the relative degree: how many times the output must be differentiated before the input appears. If the relative degree equals the system order, input-state linearization is complete. If it is less, the remaining internal states form the zero dynamics, which are not linearized and must be stable on their own. Unstable zero dynamics, the nonlinear analog of right-half-plane zeros, block the method, so it is restricted to minimum-phase systems.
Practical caveats
Exact cancellation requires an exact model; any mismatch leaves residual nonlinearity, so feedback linearization can be fragile without robustifying additions such as sliding-mode or Lyapunov redesign terms. It can also demand large control effort to cancel strong nonlinearities, and it discards nonlinearities that might have been helpful, which backstepping would keep.
- Exactly cancels nonlinearities via coordinates and feedback
- Relative degree determines how much is linearizable
- Unlinearized states form the zero dynamics, must be stable
- Requires an accurate model; sensitive to mismatch
Input-output linearization linearizes the map from input to a chosen output; input-state linearization linearizes the full state dynamics. The choice depends on whether the goal is output tracking or full-state control, and on the stability of the resulting internal dynamics.
For a design-stage plant with well-modeled, minimum-phase nonlinear dynamics, feedback linearization would expose a linear structure for tracking design, evaluated in simulation. It is a foundational tool of geometric nonlinear control.