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

Sum-of-Squares (SOS) optimization allows algorithmic search for polynomial Lyapunov functions and robust controllers. Toolboxes like SOSTOOLS and nonlinear control design via SOS are revolutionizing the field.

Advantages:

Limitations:

MPC solves an online optimization problem over a finite horizon. However, without care, it can destabilize nonlinear systems. The robust solution: add a Lyapunov-based stability constraint. At each step, enforce (V(\mathbfx_k+1) \leq V(\mathbfx_k) - \alpha V(\mathbfx_k)). This Lyapunov-based MPC (LMPC) guarantees closed-loop stability even with model mismatch, provided the terminal cost is a CLF. Advantages:

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.

Drug delivery (e.g., insulin pumps for diabetes) is highly nonlinear and patient-specific. Robust model predictive control (MPC) combined with Lyapunov techniques enforces state constraints (e.g., safe glucose levels) while rejecting meal disturbances.

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): Limitations: MPC solves an online optimization problem over

[ \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) + \sqrta(\mathbfx)^2 + b(\mathbfx)^T b(\mathbfx) b(\mathbfx) & \textif b(\mathbfx) \neq 0 \ 0 & \textotherwise \endcases ] \boldsymbol\theta(t)) + \boldsymbol\Delta(\mathbfx

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