Chapter 2 Basic Nonlinear Control Methods Overview This chapter gives a survey of the basic nonlinear control methods. First, the main classes of nonlinear systems are summarized, then some examples are presented how to find the dynamic model of simple systems and an overview of the stability methods is given. It is followed by an introduction to the concepts of some often used nonlinear control methods like input/output linearization, flatness control, backstepping and receding horizon control. 2.1 Nonlinear System Classes At the interface of a nonlinear system Σ , we can observe input and output signals. We can denote the input signal by u(·) and its value at time moment t by u(t). Simi- larly, y(·) is the output signal and y(t) is its value at time moment t . We can assume that the signals are elements of functions spaces which are usually linear spaces hav- ing an appropriate norm and some topological properties (convergence of Cauchy sequences of functions, existence of scalar product etc.). The chosen function space (Banach space, Hilbert space etc.) may depend on the control problem to be solved. The state x of the nonlinear dynamic system is an abstract information which represents the entire past signals of the system observed at the interface, that is, x(τ) = x represents in abstract form all the signals u(σ ), y(σ ), ∀σ ≤ τ . The system can be characterized by the state transition function x(t) = φ(t,τ,x,u(·)) and the output mapping y(t) = g(t,x(t),u(t)) where t is the actual time, τ and x are the initial time and initial state respectively, and for causal systems u(·) denotes the input signal between τ and t . Specifically, for linear systems the transients can be written as x(t) = Φ(t,τ)x + Θ(t,τ)u(·) and y(t) = C(t)x(t) + D(t)u(t). The state transition for linear systems can be thought as the superposition of the initial state transient and the result of the input excitation. B. Lantos, L. Márton, Nonlinear Control of Vehicles and Robots, Advances in Industrial Control, DOI 10.1007/978-1-84996-122-6_2, © Springer-Verlag London Limited 2011 11