Lfh(x)+Lgh(x)u≥−αh(h(x))(Safety/CBF)cap L sub f h of x plus cap L sub g h of x u is greater than or equal to negative alpha sub h open paren h of x close paren space (Safety/CBF)
(known as a ) such that its time derivative
along the trajectories of the system is negative semi-definite ( This special class is called a , written
: Designed as a primary text or summary of recent results in control theory. Researchers
The backbone of nonlinear control design is the . While linear systems can be analyzed using eigenvalues, nonlinear systems require more sophisticated methods. Lyapunov Direct Method (Second Method) This special class is called a
Developing state-space techniques to handle bounded uncertainties and disturbances in nonlinear systems. Control Design Methods:
𝜕V𝜕xf(x)+12𝜕V𝜕x[1γ2k(x)k(x)T−g(x)g(x)T]𝜕VT𝜕x+12q(x)=0the fraction with numerator partial cap V and denominator partial x end-fraction f of x plus one-half the fraction with numerator partial cap V and denominator partial x end-fraction open bracket the fraction with numerator 1 and denominator gamma squared end-fraction k open paren x close paren k open paren x close paren to the cap T-th power minus g of x g of x to the cap T-th power close bracket the fraction with numerator partial cap V to the cap T-th power and denominator partial x end-fraction plus one-half q open paren x close paren equals 0 is the disturbance entry matrix and and actuator constraints.
To design a controller, we must first establish a mathematical description of the plant. In the state-space paradigm, a continuous-time nonlinear system is generally expressed as a set of first-order differential equations:
Safety-Critical Control via Control Barrier Functions (CBFs)
For many physical systems, the control input enters linearly. This special class is called a , written as:
Aircraft and spacecraft represent perhaps the most demanding control applications. Modern fighter jets operate across a wide flight envelope where nonlinear effects—including angle-of-attack dynamics, thrust vectoring, and aerodynamic parameter variations—dominate. Unmanned aerial vehicles (UAVs) must perform complex maneuvers in the presence of wind gusts, sensor noise, and actuator constraints.