Lyapunov Redesign
Lyapunov redesign adds a robustifying term to a nominal controller so a known Lyapunov function keeps decreasing despite matched uncertainty.
Building on a nominal design
Suppose a nominal controller stabilizes a system and you have a Lyapunov function V proving it. Now add bounded uncertainty that enters through the same channel as the control input, so-called matched uncertainty. Lyapunov redesign augments the nominal law with an extra term chosen so that the derivative of V remains negative even in the worst case allowed by the uncertainty bound.
The robustifying term
The extra term typically points opposite to the gradient of V projected onto the input channel, scaled by the uncertainty bound. If the term uses the sign of that projection, it is discontinuous and produces a sliding mode, with chattering. A continuous approximation, saturating the sign near zero, removes chattering at the price of a small residual set to which the state converges rather than the exact origin.
What it guarantees
With the discontinuous term the origin is rendered uniformly asymptotically stable for all admissible uncertainties. With the continuous approximation the state is guaranteed to reach and stay in a small neighborhood whose size is set by the smoothing parameter, called uniform ultimate boundedness. Only matched uncertainty is handled cleanly; unmatched uncertainty requires backstepping or other structural methods.
- Start from a nominal design with a known Lyapunov function
- Add a term keeping V decreasing under worst-case uncertainty
- Discontinuous term gives sliding mode and chattering
- Continuous version gives ultimate boundedness, no chattering
Lyapunov redesign is a general recipe that unifies several robust nonlinear methods. It makes explicit the price of robustness: either accept chattering for exact regulation, or accept a bounded steady-state region for smooth control.
It sits alongside sliding-mode control and adaptive methods as a way to handle uncertainty without a precise model, relying only on a bound. For design-stage nonlinear loops, it provides certified robustness from a nominal Lyapunov certificate.