Consider the nonlinear system

where ƒ(x, u) is continuously differentiable in a domain D_xx D_u ⊂ ⁿ x ^m. First, we consider the linearization around an equilibrium point x_e, which for notational simplicity is assumed to be the origin; i.e., x = 0, u = 0. Then we consider the linearization around a nominal trajectory x*(t). Finally, we describe the concept of gain scheduling, which is a feedback control technique based on linearization around multiple operating points.

The main idea behind linearization is to approximate the nonlinear system in (5.1) by a linear model of the form

where A, B are matrices of dimension n x n and n x m, respectively. Typically, the linear model is an accurate approximation to the nonlinear system only in a neighborhood of the point around which the linearization took place. This is illustrated in Figure 5.1, which depicts linearization around x = 0. As shown in the diagram, the linearized model Ax is a good approximation of ƒ(x) for x close to zero; however, if x(t) moves significantly away from the equilibrium point x = 0, then the linear approximation is inaccurate. As a consequence, a linear control law that was designed based on the linear approximation may very well be unsuitable once the trajectory moves away from the equilibrium, possibly due to modeling errors or disturbances.

The term small-signal linearization is used to characterize the fact that the linear model is close to the real nonlinear system if the system trajectory x(t) remains close to the equilibrium point x_eor to the nominal trajectory x*(t). Therefore, for sufficiently small signals x(t) x_e, the linearized system is an accurate approximation of the nonlinear system. The term "small-signal" linearization also distinguishes this type of linearization from feedback linearization, which will be studied in the next section.

In general, feedback control techniques based on the linear model work well when applied to the nonlinear system if the uncertainty of the system is small, thus allowing the feedback controller to keep the trajectory close to the equilibrium point x_e. Obviously, linear controllers derived based on small-signal linearization have good closed-loop performance in cases where the system nonlinearities are not dominant or they do not have a destabilizing

Figure 5.1: Diagram to illustrate small-signal linearization around x = 0.

effect. For example, stabilization of the origin, the nonlinearity of the scalar system

has a stabilizing effect, thus if the control law u = 2x is used then for the resulting closed-loop systemthe origin is asymptotically stable even though the nonlinear term x³has not been removed by the control law.