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Chapter 6 Feedback Linearization
Feedback linearization is an approach to nonlinear control design which has attracted a great deal of research interest in recent years. The central idea of the approach is to algebraically transform a nonlinear system dynamics into a (fully or partly) linear one, so that linear control techniques can be applied. This differs entirely from conventional linearization (i.e., Jacobian linearization, as in section 3.3) in that feedback linearization is achieved by exact state transformations and feedback, rather than by linear approximations of the dynamics. The idea of simplifying the form of a system's dynamics by choosing a different state representation is not entirely unfamiliar. In mechanics, for instance, it is well known that the form and complexity of a system model depend considerably on the choice of reference frames or coordinate systems. Feedback linearization techniques can be viewed as ways of transforming original system models into equivalent models of a simpler form. Thus, they can also be used in the development of robust or adaptive nonlinear controllers, as discussed in chapters 7 and 8. Feedback linearization has been used successfully to address some practical control problems. These include the control of helicopters, high performance aircraft, industrial robots, and biomedical devices. More applications of the methodology are being developed in industry. However, there are also a number of important shortcomings and limitations associated with the feedback linearization approach. Such problems are still very much topics of current research.
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208
Feedback Linearization
Chap. 6
This chapter provides a description of feedback linearization, including what it is, how to use it for control design and what its limitations are. In section 6.1, the basic concepts of feedback linearization are described intuitively and illustrated with simple examples. Section 6.2 introduces mathematical tools from differential geometry which are useful to generalize these concepts to a broad class of nonlinear systems. Sections 6.3 and 6.4 describe feedback linearization theory for SISO systems, and section 6.5 extends the methodology to MIMO systems.
6.1 Intuitive Concepts This section describes the basic concepts of feedback linearization intuitively, using simple examples. The following sections will formalize these concepts for more general nonlinear systems.
6.1.1 Feedback Linearization And The Canonical Form In its simplest form, feedback linearization amounts to canceling the nonlinearities in a nonlinear system so that the closed-loop dynamics is in a linear form. This very simple idea is demonstrated in the following example. Example 6.1: Controlling the fluid level in a tank Consider the control of the level h of fluid in a tank (Figure 6.1) to a specified level hd. control input is the flow u into the tank, and the initial level is ho.
The
output flow
Figure 6.1 : Fluid level control in a tank The dynamic model of the tank is (6.1)
Sect. 6.1
Intuitive Concepts
209
where A(h) is the cross section of the tank and a is the cross section of the outlet pipe. If the initial level ha is quite different from the desired level hd, the control of h involves a nonlinear regulation problem. The dynamics (6.1) can be rewritten as A(h)'h =
u-a-\lgh
If «(/) is chosen as u(t) = a^2gh +A(h)v
(6.2)
with v being an "equivalent input" to be specified, the resulting dynamics is linear
A=v Choosing v as v = -ah
(6.3)
with h = h(t) - hd being the level error, and a being a strictly positive constant, the resulting closed loop dynamics is h + ah = 0 This implies that h(t) —> 0 as t —* °°. determined by the nonlinear control law u(t) = a^2gh -A(h)ah
(6.4) Based on (6.2) and (6.3), the actual input flow is
(6.5)
Note that, in the control law (6.5), the first part on the right-hand side is used to provide the output flow a~^2gh
, while the second part is used to raise the fluid level according to the the
desired linear dynamics (6.4). Similarly, if the desired level is a known time-varying function hd(t), the equivalent input v can be chosen as v = hd(t)-ah so as to still yield H(t) -> 0 as / -» ° ° .
•
The idea of feedback linearization, i.e., of canceling the nonlinearities and imposing a desired linear dynamics, can be simply applied to a class of nonlinear systems described by the so-called companion form, or controllability canonical form. A system is said to be in companion form if its dynamics is represented by xW =/(x) + b(x) u
(6.6)
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Chap. 6
Feedback Linearization
where u is the scalar control input, x is the scalar output of interest, x = [x,x, ... ,x(n~l}]T is the state vector, and /(x) and b(x) are nonlinear functions of the states. This form is unique in the fact that, although derivatives of x appear in this equation, no derivative of the input u is present. Note that, in state-space representation, (6.6) can be written x
\
X
2
d df
x
n-\
f(x) + b(x)u For systems which can be expressed in the controllability canonical form, using the control input (assuming b to be non-zero)
u [v f]
=l -
(6.7)
we can cancel the nonlinearities and obtain the simple input-output relation (multipleintegrator form) x(n)
=
v
Thus, the control law
with the ki chosen so that the polynomial pn + kn_\p"~l + .... + k0 has all its roots strictly in the left-half complex plane, leads to the exponentially stable dynamics
which implies that x(t) —> 0. For tasks involving the tracking of a desired output Xj(t), the control law v = xdW - koe - k2e - .... - kn_{ e^-V
(6.8)
(where e{t) = x(t) - xd{t) is the tracking error) leads to exponentially convergent tracking. Note that similar results would be obtained if the scalar x was replaced by a vector and the scalar b by an invertible square matrix. One interesting application of the above control design idea is in robotics. The following example studies control design for a two-link robot. Design for more general robots is similar and will be discussed in chapter 9.
Intuitive Concepts
Sect. 6.1
211
Example 6.2: Feedback linearization of a two-link robot Figure 6.2 provides the physical model of a two-link robot, with each joint equipped with a motor for providing input torque, an encoder for measuring joint position, and a tachometer for measuring joint velocity. The objective of the control design is to make the joint positions q s and q2 follow desired position histories q^(t) and q^t) , which are specified by the motion planning system of the robot. Such tracking control problems arise when a robot hand is required to move along a specified path, e.g., to draw circles.
Figure 6.2 : A two-link robot Using the well-known Lagrangian equations in classical dynamics, one can easily show that the dynamic equations of the robot is
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_J_
-hqx-hq2
~hq2 hqx
L
S\ 82
= [
