nonlinear in some operating regimes, there are often regimes in which the system
dynamics are linear. For example, the behavior of an operational amplifier or the lift-vs-
force dynamics of aerodynamic bodies can be described by linear models, within a certain
limited operating range of inputs. For such a system, you can perform an experiment (or a
simulation) that excites the system only in its linear range of behavior and collect the
input/output data. You can then use the data to estimate a linear plant model, and design
a PID controller for the linear model.In other cases, the effects of nonlinearities are small. In such a case, a linear model can
provide a good approximation, such that the nonlinear deviations are treated as
disturbances. Such approximations depend heavily on the input profile, the amplitude and
frequency content of the excitation signal.Linear models often describe the deviation of the response of a system from some
equilibrium point, due to small perturbing inputs. Consider a nonlinear system whose
output, y(t), follows a prescribed trajectory in response to a known input, u(t). The
dynamics are described by dx(t)/dt = f(x, u), y = g(x,u). Here, x is a vector of internal
states of the system, and y is the vector of output variables. The functions f and g, which
can be nonlinear, are the mathematical descriptions of the system and measurement
dynamics. Suppose that when the system is at an equilibrium condition, a small
perturbation to the input, Δu, leads to a small perturbation in the output, Δy:Δx ̇=
∂f
∂x
Δx+
∂f
∂u
Δu,Δy=
∂g
∂xΔx+∂g
∂uΔu.For example, consider the system of the following Simulink block diagram:7 PID Controller Tuning