Fortunately, the class of systems described by (5.20) does not include all the possible systems the are feedback linearizable. The reason is that a large number of systems are not immediately in the form described by (5.20), but they can be put into that form by a nonlinear change of coordinates, or as it is sometimes called, state transformations. In this section, we attempt to make the concept of coordinate transformation intuitively understandable without going into all the mathematical details that are sometimes associated with it.

Since we are dealing with nonlinear systems, we are interested in nonlinear state transformations. A nonlinear state transformation is a natural extension of the same concept from linear systems. For example, consider the linear input/output system

where u ∈ ^m is the input, y ∈ ^p is the output and x ∈ ⁿis the state. The above system can be transformed to a new state coordinate system z = Tx, where T is an invertible matrix. In the new z-coordinate, the system is described by

Clearly, from an input/output (u → y) viewpoint, the two systems Σ_x, Σ_z are exactly the same. As discussed in basic control courses and linear system theory textbooks [10,19,39], state transformations can be useful for putting the system into a new coordinate framework which can make the control design and analysis more convenient.

In the case of nonlinear state transformations, we have z – T(x), where T : ⁿ→ ⁿ is a function which is required to be a diffeomorphism. This means that T is smooth and its inverse T ^–1 exists and is also smooth. It is important for the reader to distinguish between a local diffeomorphism, where T is defined over a region Ω ⊂ ⁿ, and a global diffeomorphism, which is defined over the whole space ⁿ.

In the special case of a linear transformations, a diffeomorphism is equivalent to the matrix (which represents the linear operator relative to some basis) being invertible (i.e., non-zero determinant). For nonlinear transformations, one can check whether a function is a diffeomorphism by attempting to find a smooth inverse function T ^–1 such that x = T^–1(z). In cases of complex multivariable transformations it may be difficult to derive such an inverse function. In these cases, one can show local existence of a diffeomorphism by using Lemma 5.2.1, which follows from the well-known implicit function theorem.

Lemma 5.2.1 Let T(x) be a smooth function defined in a region Ω ⊂ ⁿ. If the Jacobian matrix

is nonsingular at a point x₀ ∈ Ω, then T(x) is a local diffeomorphism in a subregion of Ω.

Once a diffeomorphism T(x) is defined then it is possible to follow a similar procedure as for linear systems to derive the model appropriate relative to the new set of coordinates z = T(x). Consider the following affine nonlinear dynamical system:

where ƒ_x: Ω_x → ⁿ, G_x: Ω_x^{n x m}and h_x : Ω_x^pare smooth functions in a region Ω_x⊂ ⁿ. The above system can be transformed to a new state coordinate system z = T(x), where T is a diffeomorphism. In the new z-coordinate, the system is described by

It is important to note that, while in linear change of coordinates the transformation is always global (i.e., T it is a global diffeomorphism), for nonlinear change of coordinates it is often the case that the transformation is local.

Following the development of the concept of a diffeomorphism, we can now define the class of feedback linearizable systems. A nonlinear system

is said to be input-state feedback linearizable if there exists a diffeomorphism z = T(x), with T(0) = 0, such that

where (A, B) is a controllable pair and β(z) is an invertible matrix for all z in a domain of interest D_z⊂ ⁿ.

Therefore, we see that the class of feedback linearizable systems includes not only systems described by (5.20), but also systems that can be transformed into that form by a nonlinear state transformation. Determining if a given nonlinear system is feedback linearizable and what is an appropriate diffeomorphism are not obvious issues, and in fact they can be extremely difficult since in general they involve solving a set of partial differential equations.

Given a nonlinear system (5.25), consider a diffeomorphism z – T(x). In the z-coordinates we have

For feedback linearizable systems, (5.27) needs to be of the form

Therefore, the diffeomorphism T(x) that we are looking for needs to satisfy

Hence, we conclude that that for a diffeomorphism to be able to transform (5.25) into (5.26), it needs to satisfy the partial differential equations (5.28)-(5.29) for some α(•) and β(•). Whether a given system belongs to the class of feedback linearizable systems or not can be determined by checking two types of necessary and sufficient conditions: (i) controllability condition and (ii) an involutivity condition [121, 134, 249]. The derivation of this result, while interesting from a mathematical viewpoint, is beyond the scope of this book.

■ EXAMPLE 5.4

Consider a model of a single-link manipulator with flexible joints, which is described by

where J₁, J₂, M, g, L, k are known constants. The system can be written in state-space form by defining, Thus, we obtain

Consider the following diffeomorphism z = T(x)

Proving that (5.30) is indeed a diffeomorphism is left as an exercise (see Exercise 5.8). The dynamics of the system in the z-coordinates are given by

Therefore, if we choose the control law u as

we obtain the following set of linear equations