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 using some other method. Because of this recursive structure, the designer can start the design process at the known-stable system and 'back out' new controllers that progressively stabilize each outer subsystem. The process terminates when the final external control is reached. Hence, this process is known as backstepping. { x ˙ = f x ( x ) + g x ( x ) z 1 z ˙ 1 = u 1 {displaystyle {egin{cases}{dot {mathbf {x} }}=f_{x}(mathbf {x} )+g_{x}(mathbf {x} )z_{1}\{dot {z}}_{1}=u_{1}end{cases}}}     ( 1) V 1 ( x , z 1 ) ≜ V x ( x ) + 1 2 ( z 1 − u x ( x ) ) 2 {displaystyle V_{1}(mathbf {x} ,z_{1}) riangleq V_{x}(mathbf {x} )+{frac {1}{2}}(z_{1}-u_{x}(mathbf {x} ))^{2}}     ( 2) u 1 ( x , z 1 ) = v 1 + u ˙ x ⏟ By definition of  v 1 = − ∂ V x ∂ x g x ( x ) − k 1 ( z 1 − u x ( x ) ⏟ e 1 ) ⏞ v 1 + ∂ u x ∂ x ( f x ( x ) + g x ( x ) z 1 ⏟ x ˙  (i.e.,  d ⁡ x d ⁡ t ) ) ⏞ u ˙ x  (i.e.,  d ⁡ u x d ⁡ t ) {displaystyle underbrace {u_{1}(mathbf {x} ,z_{1})=v_{1}+{dot {u}}_{x}} _{{ ext{By definition of }}v_{1}}=overbrace {-{frac {partial V_{x}}{partial mathbf {x} }}g_{x}(mathbf {x} )-k_{1}(underbrace {z_{1}-u_{x}(mathbf {x} )} _{e_{1}})} ^{v_{1}},+,overbrace {{frac {partial u_{x}}{partial mathbf {x} }}(underbrace {f_{x}(mathbf {x} )+g_{x}(mathbf {x} )z_{1}} _{{dot {mathbf {x} }}{ ext{ (i.e., }}{frac {operatorname {d} mathbf {x} }{operatorname {d} t}}{ ext{)}}})} ^{{dot {u}}_{x}{ ext{ (i.e., }}{frac {operatorname {d} u_{x}}{operatorname {d} t}}{ ext{)}}}}     ( 3) { x ˙ = f x ( x ) + g x ( x ) z 1 z ˙ 1 = z 2 z ˙ 2 = u 2 {displaystyle {egin{cases}{dot {mathbf {x} }}=f_{x}(mathbf {x} )+g_{x}(mathbf {x} )z_{1}\{dot {z}}_{1}=z_{2}\{dot {z}}_{2}=u_{2}end{cases}}}     ( 4) { y ˙ = f y ( y ) + g y ( y ) z 2 ( where this  y  subsystem is stabilized by  z 2 = u 1 ( x , z 1 )  ) z ˙ 2 = u 2 . {displaystyle {egin{cases}{dot {mathbf {y} }}=f_{y}(mathbf {y} )+g_{y}(mathbf {y} )z_{2}&quad { ext{( where this }}mathbf {y} { ext{ subsystem is stabilized by }}z_{2}=u_{1}(mathbf {x} ,z_{1}){ ext{ )}}\{dot {z}}_{2}=u_{2}.end{cases}}}     ( 5) { x ˙ = f x ( x ) + g x ( x ) z 1 z ˙ 1 = f 1 ( x , z 1 ) + g 1 ( x , z 1 ) u 1 {displaystyle {egin{cases}{dot {mathbf {x} }}=f_{x}(mathbf {x} )+g_{x}(mathbf {x} )z_{1}\{dot {z}}_{1}=f_{1}(mathbf {x} ,z_{1})+g_{1}(mathbf {x} ,z_{1})u_{1}end{cases}}}     ( 6) u 1 ( x , z 1 ) = 1 g 1 ( x , z 1 ) ( − ∂ V x ∂ x g x ( x ) − k 1 ( z 1 − u x ( x ) ) + ∂ u x ∂ x ( f x ( x ) + g x ( x ) z 1 ) ⏞ u a 1 ( x , z 1 ) − f 1 ( x , z 1 ) ) {displaystyle u_{1}(mathbf {x} ,z_{1})={frac {1}{g_{1}(mathbf {x} ,z_{1})}}left(overbrace {-{frac {partial V_{x}}{partial mathbf {x} }}g_{x}(mathbf {x} )-k_{1}(z_{1}-u_{x}(mathbf {x} ))+{frac {partial u_{x}}{partial mathbf {x} }}(f_{x}(mathbf {x} )+g_{x}(mathbf {x} )z_{1})} ^{u_{a1}(mathbf {x} ,z_{1})},-,f_{1}(mathbf {x} ,z_{1}) ight)}     (7) V 1 ( x , z 1 ) = V x ( x ) + 1 2 ( z 1 − u x ( x ) ) 2 {displaystyle V_{1}(mathbf {x} ,z_{1})=V_{x}(mathbf {x} )+{frac {1}{2}}(z_{1}-u_{x}(mathbf {x} ))^{2}}     (8) 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 using some other method. Because of this recursive structure, the designer can start the design process at the known-stable system and 'back out' new controllers that progressively stabilize each outer subsystem. The process terminates when the final external control is reached. Hence, this process is known as backstepping. The backstepping approach provides a recursive method for stabilizing the origin of a system in strict-feedback form. That is, consider a system of the form