For a nominal system (\dot\mathbfx = \mathbff(\mathbfx)), the classical Lyapunov theorems provide:
The genius of Aleksandr Lyapunov (1857–1918) was to prove stability without explicitly solving differential equations. Instead, he introduced the concept of a Lyapunov function (V(\mathbfx)), which acts as a generalized energy function.
For a system (\dot\mathbfx = \mathbff(\mathbfx)) with (\mathbff(0)=0), if we can find a continuously differentiable function (V(\mathbfx)) such that:
Then the origin is stable. If (\dotV(\mathbfx) < 0) for all (\mathbfx \neq 0), then the origin is asymptotically stable. If additionally (V(\mathbfx) \to \infty) as (|\mathbfx| \to \infty) (radially unbounded), then the stability is global. Then the origin is stable
| Feature | Linear Robust Control (e.g., (H_\infty)) | Nonlinear Robust Control | | --- | --- | --- | | Model | LTI + norm-bounded uncertainty | Nonlinear + bounded disturbances | | Stability Guarantee | Global only if plant is LTI | Local or regional via Lyapunov | | Computational Load | Convex optimization (LMIs) | ODE solvers, symbolic computation | | Applicability | Near equilibrium | Large-signal, wide operating range |
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):
[ \inf_\mathbfu \left[ \frac\partial V\partial \mathbfx \left( \mathbff(\mathbfx) + \mathbfg(\mathbfx)\mathbfu \right) \right] < 0 ] If you want, I can produce a focused
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) + \sqrtb(\mathbfx)b(\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. \boldsymbol\theta(t)) + \boldsymbol\Delta(\mathbfx
If you want, I can produce a focused deliverable next (choose one):
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.