5.2.2 Higher-Order Input-State Linearization

Similar ideas can be developed for n-th order systems in the so called companion form:

The nonlinearities can be cancelled by using a feedback linearizing control law of the form

This results in a simple linear relation (n integrators in series) between v and x, given by

Therefore, we can choose v as

where e(t) = x₁(t) – y_d (t) is the tracking error. In this case, the characteristic equation for the tracking error dynamics of the closed-loop system is

Choosing the design coefficients {λ_o, λ₁, ... λ_{n – 1}} so that this characteristic equation is a Hurwitz polynomial (i.e., the roots of the polynomial are all in the left-half complex plane) implies that the closed-loop system is exponentially stable and e(t) converges to zero exponentially fast, with a rate that depends on the choice of the design coefficients.

A important question is: Can all functions in nonlinear systems be cancelled by such feedback methods?

Clearly, the extent of the designer's ability to cancel nonlinearities depends on the structural and physical limitations that are applicable. For example, if control actuators could be placed to allow control of every state independently (an unrealistic assumption in almost all practical applications), then under some conditions on the invertibility of the actuator gain g (x), we would be able to use each control signal to cancel the nonlinearities of the corresponding state. In general, however, it is not possible to cancel all nonlinearities by feedback linearization methods. To achieve such nonlinearity cancellation, certain structural properties in the nonlinear system must be satisfied.

A first cut at the class of feedback linearizable systems is nonlinear systems described by

where u is a m-dimensional control input, x is an n-dimensional state vector, A is an n × n matrix, B is an n × m matrix, and the pair (A, B) is controllable. The nonlinearities are contained in the functions α : ^nmand β : ^{nm × m}, which are defined on an appropriate domain of interest, with the matrix β(x) assumed to be nonsingular for every x in the domain of interest and the symbol β ^-1 denotes an inverse matrix. Systems described by (5.20) can be linearized by using a state feedback of the form

which results in

For stabilization, a state feedback v = Kx can be designed such that the closed-loop system is asymptotically stable. This is achieved by selecting K such that all the eigenvalues of A + BK are in the left-half complex plane. A similar design procedure, based on linear control designed methods, can be used to select v for tracking problems. The reader will undoubtedly notice that the class of systems described by (5.20) is significantly more general than the class of nonlinear systems in companion form (5.19). The class of feedback linearizable systems is actually even larger than the systems described by (5.20) since it includes nonlinear systems that can be transformed to (5.20) by a coordinate transformation. This topic is discussed in detail in the next subsection.

If a nonlinear system is not feedback linearizable, it does not imply that it cannot be controlled. There are several classes of nonlinear systems that cannot be put into the standard form for feedback linearizable systems, but they can be controlled by other methods.

Feedback linearization, although a very useful tool with a beautiful mathematical theory for dealing with nonlinear systems, has some serious drawbacks in practical applications. Two of these drawbacks are discussed below:

• Feedback linearization may not be the most efficient way of controlling a nonlinear system. To illustrate this concept consider the (frequently used) simple system
  
  
  
  For stabilization around x = 0, a feedback linearizing controller would cancel the term x³. However, this is a "stable" term so there is not real need to cancel it. Instead, a simple linear feedback control law of the form u = –x, could achieve similar results without a large control effort, as compared to linearizing feedback controller of the form u = –x + x³. The reader will undoubtedly note that if the initial state x(0) is far away from zero, then the feedback linearizing controller will require significantly larger control effort than a linear control law. The bottom line is that, in this case, the controller is working hard to cancel a nonlinearity that is actually a stable term helping the control effort. The concept of canceling useful nonlinearities is also present in higher dimensional systems, however it becomes less evident due to the complexity of the problem. Note that this issue is less important when the objective is tracking. In the above example, when the objective is to cause x to track y_dthen the x³term would have to be addressed, e.g.,
  
  
  
  
• Feedback linearization relies heavily on the exact cancellation of nonlinearities. In practice, the nonlinear terms of a dynamical system are not known exactly, therefore exact cancellation may not be possible. By their nature, linearization methods are not "robust" with respect to modeling or other uncertainties. For example, consider a feedback linearizable system of the form
  
  
  
  
  where in the actual system, ε > 0. However, because of lack of knowledge about the value of ε, the designer had assumed that ε = 0, thus designing a stabilizing control law of the form u = –x – x². In this case, the closed-loop system is given by
  
  
  which is unstable if
  
  
  
  Moreover, if x(0) > ε^–1/3, then x(t) → ∞ in finite time – this is called finite escape time. For tracking control, the issue of modeling errors can be even more critical, since the signal y_d may cause the state to move into the portion of the state space where the model error is significant (e.g., y_d >ε^–1/3 in the example of this paragraph). Methods to accommodate modeling error are presented in Section 5.4 using bounding techniques, in Section 5.5 using adaptive techniques and in Chapters 6 and 7 using adaptive approximation methods.