NONLINEAR SYSTEMS

1) Introduction. Nonlinear phenomena and mathematical models, applications to automation and robotics. Examples. Nonlinear state models: existence and uniqueness of solutions. Second-order dynamical systems: qualitative behavior of linear systems, phase diagrams, multiple equilibria, limit cycles. Useful mathematics for nonlinear systems. Modeling examples: kinematics of wheeled vehicles, magnetic levitators, overhead cranes. [11 hours]
2) Autonomous systems. Stability theory: the direct method of Lyapunov, Lyapunov functions and the variable gradient method. The region of attraction of an equilibrium state. Global asymptotic stability: Barbashin-Krasovkii theorem. Instability: Chetaev theorem. The algebraic Lyapunov equation and the indirect method. The invariance principle: LaSalle theorem. Stability and attractiveness of state sets. Limit cycles in feedback systems: method of the describing function. [14 hours]
3) Nonautonomous systems. Stability of state motions. Class K and class KL (comparison) functions. The direct method for the uniform asymptotic stability. Input-to-state stability. The direct and indirect method for exponential stability. Converse theorems in stability theory. [7 hours]
4) Nonlinear control. The stabilization problem. State-input feedback methods: control Lyapunov functions, integrator backstepping. Relative degree and normal form of a scalar affine control system. Input-output linearization by state-input feedback (feedback linearization). Zero dynamics and minimum-phase systems. Application to stabilization. Regulation of nonlinear scalar systems: integral control. Input-output inversion-based feedforward control. Stable inversion for nonminimum-phase systems: closed-form solutions for linear systems and iterative method for nonlinear systems. Feedforward-feedback control schemes. [16 hours]