All physical systems are nonlinear to some extent. Several inherent properties of linear systems which greatly simplify the solution for this class of systems, are not valid for nonlinear systems (Shinner, 1998). The fact that nonlinear systems do not have these properties further complicates their analysis. Moreover, nonlinearities usually appear multiplied with physical constants, often poorly known or dependent on the slowly changing environment, thereby increasing the complexities. Therefore, it is important that one acquires a facility for analyzing control systems with varying degrees of nonlinearity. This section introduces three nonlinear control methods for tracking purposes. To maintain generality, we consider a general dynamic model of the form
that can represent the dynamic model of numerous mechanical systems such as robotic vehicles, robot manipulators, etc, where H(q) is an n xn matrix, representing mass matrix or inertia matrix (including added mass for underwater vehicles), C(q, q) represents the matrix of Coriolis and centripetal terms (including added mass for underwater vehicles), and G(q) is the vector of gravitational forces and moments. For the case of underwater vehicles, which is the main concern of this chapter, the term C(q, q) will also represent the hydrodynamic damping and lift matrix. The methods given in this section, will be applied to an underwater vehicle model in section 3.
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