Feedback Linearization

Using state feedback to exactly cancel nonlinear terms, turning the system into a linear one before designing a controller for it.

Feedback linearization is a fundamental method in nonlinear control: design a control law u = a(x) + b(x)·v, where a and b are computed from the system model to exactly cancel the nonlinear terms; combined with a change of coordinates, this turns the relationship from the new input v to the output into something simple and linear (commonly a chain of integrators), after which PD control, pole placement, or other linear methods can design v. It differs from the common approach of ‘Taylor-expanding around an operating point’ (Jacobian linearization): that is only an approximation valid near the operating point, while feedback linearization is an exact transformation, valid over a much larger range — but only if the model is accurate; with model error, the cancellation is imperfect and robustness suffers. Computed torque control for robot arms is a special case of this method. When only the output is made linear, whatever internal dynamics remain hidden are called the zero dynamics, and these must be stable; hybrid zero dynamics (HZD) control for bipedal walking is built on exactly this kind of input-output linearization.

ExampleArm dynamics M(q)q̈ + C(q, q̇)q̇ + g(q) = τ (M the mass matrix, the C term Coriolis and centrifugal forces, g gravity, τ joint torque). Substituting τ = M(q)v + C(q, q̇)q̇ + g(q) gives q̈ = v — each joint becomes an independent double integrator; then setting v = q̈_d + K_d(q̇_d − q̇) + K_p(q_d − q) makes the tracking error converge as a linear second-order system.

Also called

Exact Linearization, Input-Output Linearization

Related

Computed Torque Control · Inverse Dynamics · Hybrid Zero Dynamics · Operational Space Control · Mass Matrix · Lyapunov Stability

Sources

Wikipedia: Feedback linearization