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19. Introduction to Nonlinear Control

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.

• 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 directly-applicable 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 pre-compensated signals (caution, however with singular points such as division by zero, etc, for particular signal values).
     
• Non-smooth 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

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