In control theory, dynamical systems are in strict-feedback form when they can be expressed as

where

  • with ,
  • are scalars,
  • is a scalar input to the system,
  • vanish at the origin (i.e., ),
  • are nonzero over the domain of interest (i.e., for ).

Here, strict feedback refers to the fact that the nonlinear functions and in the equation only depend on states that are fed back to that subsystem.[1][page needed] That is, the system has a kind of lower triangular form.

Stabilization

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Systems in strict-feedback form can be stabilized by recursive application of backstepping.[1][page needed] That is,

  1. It is given that the system
     
    is already stabilized to the origin by some control   where  . That is, choice of   to stabilize this system must occur using some other method. It is also assumed that a Lyapunov function   for this stable subsystem is known.
  2. A control   is designed so that the system
     
    is stabilized so that   follows the desired   control. The control design is based on the augmented Lyapunov function candidate
     
    The control   can be picked to bound   away from zero.
  3. A control   is designed so that the system
     
    is stabilized so that   follows the desired   control. The control design is based on the augmented Lyapunov function candidate
     
    The control   can be picked to bound   away from zero.
  4. This process continues until the actual   is known, and
    • The real control   stabilizes   to fictitious control  .
    • The fictitious control   stabilizes   to fictitious control  .
    • The fictitious control   stabilizes   to fictitious control  .
    • ...
    • The fictitious control   stabilizes   to fictitious control  .
    • The fictitious control   stabilizes   to fictitious control  .
    • The fictitious control   stabilizes   to the origin.

This process is known as backstepping because it starts with the requirements on some internal subsystem for stability and progressively steps back out of the system, maintaining stability at each step. Because

  •   vanish at the origin for  ,
  •   are nonzero for  ,
  • the given control   has  ,

then the resulting system has an equilibrium at the origin (i.e., where  ,  ,  , ... ,  , and  ) that is globally asymptotically stable.

See also

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References

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  1. ^ a b Khalil, Hassan K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.