In control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

, dynamical systems

Dynamical systems theory

Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference...

 are in strict-feedback form when they can be expressed as


where

• with ,
• are scalar
  Scalar (mathematics)
  In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
  
  s,
• is a scalar
  Scalar (mathematics)
  In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
  
   input to the system,
• vanish
  Vanish
  Vanish may refer to:*Vanish , a toilet bowl cleaner by S.C. Johnson or a cloth stain remover by Reckitt Benckiser*"Vanish" an episode of the TV series Criss Angel Mindfreak*Vanishing, a type of magical effect...
  
   at the origin
  Origin (mathematics)
  In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
  
   (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. That is, the system has a kind of lower triangular

Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...

 form.

Stabilization

Systems in strict-feedback form can be stabilized

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

 by recursive application of backstepping

Backstepping

In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic and others for designing stabilizing controls for a special class of nonlinear dynamical systems. These systems are built from subsystems that radiate out from an irreducible subsystem that can be stabilized...

. 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

   Lyapunov function

   In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...

   
     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.