Variable structure system

A variable structure system, or VSS, is a discontinuous nonlinear system of the form x ˙ = φ ( x , t ) {\displaystyle {\dot {\mathbf {x} }}=\varphi (\mathbf {x} ,t)} where x ≜ [ x 1 , x 2 , … , x n ] T ∈ R n {\displaystyle \mathbf {x} \triangleq [x_{1},x_{2},\ldots ,x_{n}]^{\operatorname {T} }\in \mathbb {R} ^{n}} is the state vector, t ∈ R {\displaystyle t\in \mathbb {R} } is the time variable, and φ ( x , t ) ≜ [ φ 1 ( x , t ) , φ 2 ( x , t ) , … , φ n ( x , t ) ] T : R n + 1 ↦ R n {\displaystyle \varphi (\mathbf {x} ,t)\triangleq [\varphi _{1}(\mathbf {x} ,t),\varphi _{2}(\mathbf {x} ,t),\ldots ,\varphi _{n}(\mathbf {x} ,t)]^{\operatorname {T} }:\mathbb {R} ^{n+1}\mapsto \mathbb {R} ^{n}} is a piecewise continuous function. Due to the piecewise continuity of these systems, they behave like different continuous nonlinear systems in different regions of their state space.

Source: Wikipedia — Variable structure system (CC BY-SA 4.0)

Variable structure system

A variable structure system, or VSS, is a discontinuous nonlinear system of the form x ˙ = φ ( x , t ) {\displaystyle {\dot {\mathbf {x} }}=\varphi (\mathbf {x} ,t)} where x ≜ [ x 1 , x 2 , … , x n ] T ∈ R n {\displaystyle \mathbf {x} \triangleq [x_{1},x_{2},\ldots ,x_{n}]^{\operatorname {T} }\in \mathbb {R} ^{n}} is the state vector, t ∈ R {\displaystyle t\in \mathbb {R} } is the time variable, and φ ( x , t ) ≜ [ φ 1 ( x , t ) , φ 2 ( x , t ) , … , φ n ( x , t ) ] T : R n + 1 ↦ R n {\displaystyle \varphi (\mathbf {x} ,t)\triangleq [\varphi _{1}(\mathbf {x} ,t),\varphi _{2}(\mathbf {x} ,t),\ldots ,\varphi _{n}(\mathbf {x} ,t)]^{\operatorname {T} }:\mathbb {R} ^{n+1}\mapsto \mathbb {R} ^{n}} is a piecewise continuous function. Due to the piecewise continuity of these systems, they behave like different continuous nonlinear systems in different regions of their state space.

Source: Wikipedia "Variable structure system" · CC BY-SA 4.0

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