Robust Nonlinear Control Design State Space And: Lyapunov Techniques Systems Control Foundations Applications

A nonlinear system in state space form is written as:

[ \beginaligned \dot\mathbfx(t) &= \mathbff(\mathbfx(t), \mathbfu(t), \boldsymbol\theta(t)) + \boldsymbol\Delta(\mathbfx, \mathbfu, t) \ \mathbfy(t) &= \mathbfh(\mathbfx(t)) \endaligned ]

where:

Key idea: Uncertainty is often described in a structured or unstructured manner. Robust control seeks to guarantee properties (e.g., boundedness, convergence) for all possible uncertainties within a known set.

Suppose we have a nominal nonlinear system (\dot\mathbfx = \mathbff(\mathbfx) + \mathbfg(\mathbfx)\mathbfu) with a known CLF and a stabilizing control (\mathbfu_\textnom(\mathbfx)). Now add a bounded disturbance (\mathbfd(t)) and parametric uncertainty (\Delta(\mathbfx)): A nonlinear system in state space form is

[ \dot\mathbfx = \mathbff(\mathbfx) + \mathbfg(\mathbfx)\mathbfu + \Delta(\mathbfx) + \mathbfd(t) ]

The Lyapunov redesign approach introduces an extra robustifying term (\mathbfu_\textrob(\mathbfx)) such that:

[ \dotV \leq -\alpha V(\mathbfx) + \epsilon ]

This ensures ultimately bounded stability (i.e., the state converges to a ball around the origin). The robust term often takes the form of a signum or saturation function: Key idea: Uncertainty is often described in a

[ \mathbfu_\textrob = -\rho(\mathbfx) , \textsign\left( \frac\partial V\partial \mathbfx \mathbfg(\mathbfx) \right) ]

This is the essence of sliding mode control, one of the most powerful robust nonlinear methods.

A robust nonlinear control design framework using state‑space and Lyapunov methods should provide tools and methods to model nonlinear systems, analyze stability under uncertainties/disturbances, synthesize controllers that guarantee performance and robustness, and validate results analytically and via simulation.

For control systems (\dot\mathbfx = \mathbff(\mathbfx) + \mathbfg(\mathbfx)\mathbfu), a Control Lyapunov Function is a (V(\mathbfx) > 0) such that for every (\mathbfx \neq 0): analyze stability under uncertainties/disturbances

[ \inf_\mathbfu \left[ \frac\partial V\partial \mathbfx \left( \mathbff(\mathbfx) + \mathbfg(\mathbfx)\mathbfu \right) \right] < 0 ]

This means there exists a control law that can decrease (V) at every point. The famous Sontag’s formula provides a universal stabilizing controller when a CLF is known:

[ \mathbfu(\mathbfx) = \begincases -\fraca(\mathbfx) + \sqrt^4b(\mathbfx)^T b(\mathbfx) b(\mathbfx) & \textif b(\mathbfx) \neq 0 \ 0 & \textotherwise \endcases ]

where (a(\mathbfx) = L_f V(\mathbfx)) and (b(\mathbfx) = L_g V(\mathbfx)). This is a cornerstone of robust nonlinear design.