Robust Nonlinear Control Design State Space And Lyapunov Techniques Systems Control Foundations Applications [new]

Robust Nonlinear Control Design: State-Space and Lyapunov Techniques

Computational Methods & Optimization

• Sum‑of‑squares (SOS) programming to search polynomial Lyapunov functions and certify ROA
• Linear matrix inequalities (LMIs) for convex relaxations of stability/robustness conditions
• Semi‑definite programming (SDP) solvers integration
• Numerical methods for Hamilton–Jacobi PDEs for value‑function/robust control synthesis
• Trajectory optimization and direct collocation for NMPC
• Tools for certifiable numerical bounds (interval arithmetic, validated numerics)

3.2 Lyapunov Redesign: Adding Robustness to a Nominal Design

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)):

State Space Techniques

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: