In control theory, dynamical systems are in strict-feedback form when they can be expressed as { x ˙ = f 0 ( x ) + g 0 ( x ) z 1 z ˙ 1 = f 1 ( x , z 1 ) + g 1 ( x , z 1 ) z 2 z ˙ 2 = f 2 ( x , z 1 , z 2 ) + g 2 ( x , z 1 , z 2 ) z 3 ⋮ z ˙ i = f i ( x , z 1 , z 2 , … , z i − 1 , z i ) + g i ( x , z 1 , z 2 , … , z i − 1 , z i ) z i + 1 for 1 ≤ i < k − 1 ⋮ z ˙ k − 1 = f k − 1 ( x , z 1 , z 2 , … , z k − 1 ) + g k − 1 ( x , z 1 , z 2 , … , z k − 1 ) z k z ˙ k = f k ( x , z 1 , z 2 , … , z k − 1 , z k ) + g k ( x , z 1 , z 2 , … , z k − 1 , z k ) u {\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{0}(\mathbf {x} )+g_{0}(\mathbf {x} )z_{1}\\{\dot {z}}_{1}=f_{1}(\mathbf {x} ,z_{1})+g_{1}(\mathbf {x} ,z_{1})z_{2}\\{\dot {z}}_{2}=f_{2}(\mathbf {x} ,z_{1},z_{2})+g_{2}(\mathbf {x} ,z_{1},z_{2})z_{3}\\\vdots \\{\dot {z}}_{i}=f_{i}(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{i-1},z_{i})+g_{i}(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{i-1},z_{i})z_{i+1}\quad {\text{ for }}1\leq i

Property Value
dbo:abstract
• In control theory, dynamical systems are in strict-feedback form when they can be expressed as { x ˙ = f 0 ( x ) + g 0 ( x ) z 1 z ˙ 1 = f 1 ( x , z 1 ) + g 1 ( x , z 1 ) z 2 z ˙ 2 = f 2 ( x , z 1 , z 2 ) + g 2 ( x , z 1 , z 2 ) z 3 ⋮ z ˙ i = f i ( x , z 1 , z 2 , … , z i − 1 , z i ) + g i ( x , z 1 , z 2 , … , z i − 1 , z i ) z i + 1 for 1 ≤ i < k − 1 ⋮ z ˙ k − 1 = f k − 1 ( x , z 1 , z 2 , … , z k − 1 ) + g k − 1 ( x , z 1 , z 2 , … , z k − 1 ) z k z ˙ k = f k ( x , z 1 , z 2 , … , z k − 1 , z k ) + g k ( x , z 1 , z 2 , … , z k − 1 , z k ) u {\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{0}(\mathbf {x} )+g_{0}(\mathbf {x} )z_{1}\\{\dot {z}}_{1}=f_{1}(\mathbf {x} ,z_{1})+g_{1}(\mathbf {x} ,z_{1})z_{2}\\{\dot {z}}_{2}=f_{2}(\mathbf {x} ,z_{1},z_{2})+g_{2}(\mathbf {x} ,z_{1},z_{2})z_{3}\\\vdots \\{\dot {z}}_{i}=f_{i}(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{i-1},z_{i})+g_{i}(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{i-1},z_{i})z_{i+1}\quad {\text{ for }}1\leq i<k-1\\\vdots \\{\dot {z}}_{k-1}=f_{k-1}(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{k-1})+g_{k-1}(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{k-1})z_{k}\\{\dot {z}}_{k}=f_{k}(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{k-1},z_{k})+g_{k}(\mathbf {x} ,z_{1},z_{2},\dots ,z_{k-1},z_{k})u\end{cases}}} where x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} with n ≥ 1 {\displaystyle n\geq 1} , z 1 , z 2 , … , z i , … , z k − 1 , z k {\displaystyle z_{1},z_{2},\ldots ,z_{i},\ldots ,z_{k-1},z_{k}} are scalars, u {\displaystyle u} is a scalar input to the system, f 0 , f 1 , f 2 , … , f i , … , f k − 1 , f k {\displaystyle f_{0},f_{1},f_{2},\ldots ,f_{i},\ldots ,f_{k-1},f_{k}} vanish at the origin (i.e., f i ( 0 , 0 , … , 0 ) = 0 {\displaystyle f_{i}(0,0,\dots ,0)=0} ), g 1 , g 2 , … , g i , … , g k − 1 , g k {\displaystyle g_{1},g_{2},\ldots ,g_{i},\ldots ,g_{k-1},g_{k}} are nonzero over the domain of interest (i.e., g i ( x , z 1 , … , z k ) ≠ 0 {\displaystyle g_{i}(\mathbf {x} ,z_{1},\ldots ,z_{k})\neq 0} for 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} ).Here, strict feedback refers to the fact that the nonlinear functions f i {\displaystyle f_{i}} and g i {\displaystyle g_{i}} in the z ˙ i {\displaystyle {\dot {z}}_{i}} equation only depend on states x , z 1 , … , z i {\displaystyle x,z_{1},\ldots ,z_{i}} that are fed back to that subsystem. That is, the system has a kind of lower triangular form. (en)
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