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Chapter 19
19. Introduction to Nonlinear Control
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With the exception of our treatment of actuator limits, all previous
material in the book has been aimed at linear systems. This is justified
by the fact that most real world systems exhibit (near) linear behaviour
and by the significantly enhanced insights available in the linear case.
However, one occasionally meets a problem where the nonlinearities are
so important that they cannot be ignored. This chapter is intended to
give a brief introduction to nonlinear control. Our objective
is not to be comprehensive but to simply give some simple extensions of
linear strategies that might allow a designer to make a start
on a nonlinear problem. As far as possible, we will build on the linear
methods so as to maximally benefit from linear insights. We also give a
"taste" of more rigorous nonlinear theory so as to give the
reader an appreciation for this fascinating and evolving subject.
Summary
 So far, the entire book has dealt with linear system and
controllers.
 This chapter generalizes the scope to include various types of
nonlinearities:
 A number of properties that are very helpful in linear control are
not  or not directlyapplicable to the nonlinear case.
 Frequency analysis: The response to a sinusoidal signal
is not necessarily a sinusoid; therefore frequency analysis,
Bode plots, etc, cannot be directly carried over from the linear
case.
 Transfer functions: The notion of transfer functions,
poles, zeros and their respective cancellation is not directly
applicable.
 Stability becomes more involved.
 Inversion: It was highlighted in Chapter 15, on affine
structures, that whether or not the controller contains the
inverse of the model as a factor and whether or not one inverts
the model explicitly  control is fundamentally linked to the
ability to invert. Numerous nonlinear functions encountered,
however, are not invertible (such as saturations, for example).
 Superposition does not apply; that is: the effects of
two signals (such as setpoint and disturbance) acting on the
system individually cannot simply be summed (superimposed) to
determine the effect of the signals acting simultaneously on the
system.
 Commutativity does not apply.
 As a consequence, the mathematics for nonlinear control become
more involved, solutions and results are not as complete and
intuition can fail more easily than in the linear case.
 Nevertheless, nonlinearities are frequently encountered and are a
very important consideration.
 Smooth static nonlinearities at input and output
 are frequently a consequence of nonlinear actuator and sensor
characteristics
 are the easiest form of nonlinearities to compensate
 can be compensated by applying the inverse function to the
relevant signal, thus obtaining a linear system in the
precompensated signals (caution, however with singular points
such as division by zero, etc, for particular signal values).
 Nonsmooth nonlinearities cannot in general be exactly compensated
or linearized.
 The chapter applies a nonlinear generalization of the affine
parameterization of Chapter 15 to construct a controller that
generates a feedback linearizing controller if the model is smoothly
nonlinear with stable inverse
 Nonlinear stability can be investigated using a variety of
techniques. Two common strategies are
 Lyapunov methods
 Function Space methods
 Extensions of linear robustness analysis to the nonlinear case are
possible
 There also exist nonlinear sensitivity limitations which mirror
those for the linear case
