The state-space approach provides a rigorous mathematical framework for modeling complex systems. is the state vector, is the control input, and is the uncertainty.
The key takeaways for the practicing engineer:
Several foundational frameworks leverage state-space structures and Lyapunov functions to enforce robust performance despite model mismatches. 1. Feedback Linearization Lyapunov stability theory
Input-to-State Stability, introduced by Eduardo Sontag, provides a framework for analyzing how external inputs (disturbances, reference signals) affect system stability. A system is ISS if there exist functions ( \beta \in \mathcalKL ) and ( \gamma \in \mathcalK ) such that, for any initial condition ( x(0) ) and any bounded input ( u ):
Aircraft and spacecraft exhibit highly nonlinear dynamics during high-angle-of-attack maneuvers or atmospheric re-entry. Robust nonlinear control ensures stability despite varying atmospheric density and mass fluctuations. Robotics and Autonomous Vehicles 1. Feedback Linearization Input-to-State Stability
ẋ1=f1(x1)+g1(x1)x2x dot sub 1 equals f sub 1 of open paren x sub 1 close paren plus g sub 1 of open paren x sub 1 close paren x sub 2
ẋn=fn(x)+gn(x)ux dot sub n equals f sub n of x plus g sub n of x u introduced by Eduardo Sontag
represents the drift dynamics (the system's natural behavior when no control is applied), and
The authors combine concepts from set-valued analysis , Lyapunov stability theory , and game theory to develop control methods for low-order nonlinear ordinary differential equations.
by Randy A. Freeman and Petar V. Kokotović . Originally published as part of the Systems & Control: Foundations & Applications series, it remains a primary reference for engineers tackling large-signal robustness in nonlinear systems.
The theoretical foundation of nonlinear control has been translated into several practical and powerful design methodologies: