Robust Nonlinear Control Design State Space And Lyapunov Techniques Systems Control Foundations Applications (2024)
Robust Nonlinear Control Design: State-Space and Lyapunov Techniques (part of the Springer Systems & Control series) provides a unified, global framework for controlling nonlinear systems by merging Lyapunov stability theory, set-valued analysis, and game theory. The approach ensures robust stabilization against uncertainties and disturbances, utilizing methods like Input-to-State Stability (ISS) and backstepping to guarantee performance beyond linear approximations. For more information, visit Springer.
In the neon-soaked skyline of Neo-Kyoto, 2084, the "Lyapunov Towers" stood as a testament to human ambition—and its fragility. The city’s gravity-stabilization grid, governed by the archaic State Space protocols, was failing. Beneath the surface, the equations that kept the floating districts from plummeting into the sea were becoming erratic.
[ \beginaligned \dot\mathbfx(t) &= \mathbff(\mathbfx(t), \mathbfu(t), \boldsymbol\theta(t)) + \boldsymbol\Delta(\mathbfx, \mathbfu, t) \ \mathbfy(t) &= \mathbfh(\mathbfx(t)) \endaligned ] In the context of control design , this theory is inverted
8. Future Directions
- Data‑driven robust control: Learning uncertainty bounds from data while preserving Lyapunov certificates.
- Sum‑of‑squares (SOS) programming: Automated search for polynomial Lyapunov functions for robust nonlinear systems.
- Event‑triggered robust control: Reduce communication/computation while maintaining Lyapunov‑based robustness.
- Safe robust control with barrier functions: Combine Lyapunov stability with safety constraints under uncertainty.
In the context of control design, this theory is inverted. Instead of analyzing a given system, the engineer constructs the control law $u$ specifically to make $\dotV$ negative. This is known as Lyapunov-based control design (often implemented via Control Lyapunov Functions, or CLFs).
can be designed to have a "margin" that absorbs small perturbations. 3.2 Recursive Design: Backstepping In the context of control design
The text is practically self-contained and serves graduate students, researchers, and design engineers who require a deep understanding of nonlinear ordinary differential equations. If you'd like, I can:
This energy-based reasoning is the cornerstone of nonlinear design. It transforms the problem of control design into an optimization problem: finding a control law (u) that forces the derivative of the Lyapunov function to be negative. However, in the real world, achieving a mathematically perfect derivative is impossible due to uncertainties. in the real world
Here, (\mathbfx \in \mathbbR^n) is the state vector (position, velocity, pressure, flux, etc.), (\mathbfu \in \mathbbR^m) is the control input, and (\mathbfy \in \mathbbR^p) is the output. The functions (\mathbff) and (\mathbfh) are generally nonlinear and potentially time-varying.