To illustrate the main intuitive idea behind feedback linearization, we start by considering the simple scalar system

where u is the control input, y is the measured output, and the nonlinear functions ƒ, g are assumed to be known a priori. The control objective is to design a control law that generates u such that u(t) and y(t) remain bounded and y(t) tracks a desired function y_d (t). We will assume throughout that y_d (t) and all of its derivatives that are required for computing the control signal are in fact available, continuous, and bounded. Section A.4 of the appendix discusses prefiltering, which is one method to ensure the validity of this assumption. For this scalar system it is straightforward to see that, assuming that g(y) ≠ 0, the control law

where a_m > 0 is a design constant, achieves the control objective. Specifically, with the above feedback control algorithm, the tracking error e(t) = y(t) – y_d(t) satisfies Hence, the tracking error converges to zero exponentially fast from any initial condition (global stability results).

A key observation for the reader is that implementation of the feedback control algorithm (5.14) is feasible in all scenarios of desired trajectories y_donly if the function g(y) ≠ 0 for all y ∈ . Otherwise, if g(y) approaches zero then the control effort becomes large, causing saturation of the control input and possibly leading to instability. This problem, which arises due to the lack of controllability at some values of the state-space, is referred to as the stabilizability problem.

■ EXAMPLE 5.3

It is important to note that even if g(y) = 0 at a crucial part of the state-space, that does not necessarily imply that the system is uncontrollable. For example, consider the input-output system

The control law (5.14) illustrates the use of the controller for canceling nonlinearities. Specifically, as we can see from (5.14), the nonlinearities ƒ and g in the open-loop system are cancelled by the controller. This converts the system into one with linear error dynamics, for which there are known control design and analysis methods. In fact, (5.14), can be rewritten as

where (5.15) is a feedback linearizing operator that causes the closed-loop system to transform to the linear system and (5.16) is a linear stabilizing controller for the linearized tracking problem. Many other linear controllers could be selected. Even for this simple system we can extract some key observations:

• The feedback linearizing operator of (5.15) exactly linearizes the model over the domain of validity of that model. There are no approximations. This is distinct from the small signal linearization of Section 5.1, which was exact only at a single point.
  
  
• The role of the design parameter a_m> 0 is to set the time constant of the exponential convergence of the tracking error in response to initial condition errors and disturbances.
  
  
• The parameter a_m does not determine the bandwidth of the overall control system in the sense of the bandwidth of input signals y_dthat can be tracked. Note that the exponential convergence of the tracking error dynamics is independent of the input signal y_d. This is achieved by feeding forward the derivative of the input signal,. Therefore, from a theoretical perspective, the reference input tracking bandwidth of this controller is infinite. In fact, this bandwidth will be limited by physical constraints, such as the actuators, and must be accounted for in the design of the system that generates y_dand its derivatives.
  
  
• The linearization achieved by the feedback operator (5.15) requires exact knowledge of ƒ and g. The effect of model errors requires further analysis.

These comments also apply to feedback linearization when it is applied to higher order systems.

Appended Integrators. One role of integrators in control laws is to force the tracking error to zero in the presence of model error, disturbances, and input type. The required number of integrators as a function of the type of the input to be tracked is discussed in most text books on control system design, e.g., [66, 86, 140]. Integrators can have similar utility in nonlinear control applications. Integrators can be included in the control law and control design analysis by various approaches such as that discussed in Exercise 1.3 and the following.

In addition to the tracking error e(t) = y(t) – y_d (t) define

where c > 0. It is noted that e_F (t) is a linear combination of the tracking error and the integral of the tracking error that can be thought of as providing a PI controller (proportional-integral control). For implementation and analysis, the system state space model will include one appended controller state to compute the integral of the tracking error. From (5.17), we obtain ; hence, to force e_F (t) to zero, the control law (5.15) is modified to

(see also (6.35)). This control law results in . It is easy to see that if e_F (t) converges to zero then so does e(t) (notice that ).