System Modeling, part 2 Marc Claesen February 18, 2015 Marc - PowerPoint PPT Presentation

Nonlinear systems & linearization System identification (cont) System Modeling, part 2 Marc Claesen February 18, 2015 Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear systems & linearization 1 System identification (cont) 2 Grey box identification Black box identification Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Outline Nonlinear systems & linearization 1 System identification (cont) 2 Grey box identification Black box identification Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear systems In this course we focus on the linear state-space representation: � ˙ � x [ k + 1] x ( t ) = Ax ( t ) + Bu ( t ) , = Ax [ k ] + Bu [ k ] , y ( t ) = Cx ( t ) + Du ( t ) . y [ k ] = Cx [ k ] + Du [ k ] . Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear systems In this course we focus on the linear state-space representation: � ˙ � x [ k + 1] x ( t ) = Ax ( t ) + Bu ( t ) , = Ax [ k ] + Bu [ k ] , y ( t ) = Cx ( t ) + Du ( t ) . y [ k ] = Cx [ k ] + Du [ k ] . Most real life systems involve nonlinearity: � ˙ � � x ( t ) = f x ( t ) , u ( t ) , � � y ( t ) = g x ( t ) , u ( t ) , Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear systems In this course we focus on the linear state-space representation: � ˙ � x [ k + 1] x ( t ) = Ax ( t ) + Bu ( t ) , = Ax [ k ] + Bu [ k ] , y ( t ) = Cx ( t ) + Du ( t ) . y [ k ] = Cx [ k ] + Du [ k ] . Most real life systems involve nonlinearity: � ˙ � � x ( t ) = f x ( t ) , u ( t ) , � � y ( t ) = g x ( t ) , u ( t ) , where f and/or g contain some nonlinearity, such as: Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear systems In this course we focus on the linear state-space representation: � ˙ � x [ k + 1] x ( t ) = Ax ( t ) + Bu ( t ) , = Ax [ k ] + Bu [ k ] , y ( t ) = Cx ( t ) + Du ( t ) . y [ k ] = Cx [ k ] + Du [ k ] . Most real life systems involve nonlinearity: � ˙ � � x ( t ) = f x ( t ) , u ( t ) , � � y ( t ) = g x ( t ) , u ( t ) , where f and/or g contain some nonlinearity, such as: x ( t ) = Ax ( t ) + Bu ( t )+ γ u ( t ) 2 , powers : e.g. ˙ Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear systems In this course we focus on the linear state-space representation: � ˙ � x [ k + 1] x ( t ) = Ax ( t ) + Bu ( t ) , = Ax [ k ] + Bu [ k ] , y ( t ) = Cx ( t ) + Du ( t ) . y [ k ] = Cx [ k ] + Du [ k ] . Most real life systems involve nonlinearity: � ˙ � � x ( t ) = f x ( t ) , u ( t ) , � � y ( t ) = g x ( t ) , u ( t ) , where f and/or g contain some nonlinearity, such as: x ( t ) = Ax ( t ) + Bu ( t )+ γ u ( t ) 2 , powers : e.g. ˙ interactions : e.g. ˙ x ( t ) = Ax ( t ) + Bu ( t )+ γ x ( t ) u ( t ), Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear systems In this course we focus on the linear state-space representation: � ˙ � x [ k + 1] x ( t ) = Ax ( t ) + Bu ( t ) , = Ax [ k ] + Bu [ k ] , y ( t ) = Cx ( t ) + Du ( t ) . y [ k ] = Cx [ k ] + Du [ k ] . Most real life systems involve nonlinearity: � ˙ � � x ( t ) = f x ( t ) , u ( t ) , � � y ( t ) = g x ( t ) , u ( t ) , where f and/or g contain some nonlinearity, such as: x ( t ) = Ax ( t ) + Bu ( t )+ γ u ( t ) 2 , powers : e.g. ˙ interactions : e.g. ˙ x ( t ) = Ax ( t ) + Bu ( t )+ γ x ( t ) u ( t ), clipping : e.g. α ≤ x ( t ) ≤ β . Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization around equilibrium point Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization around equilibrium point Nonlinear systems have (several) equilibrium points x e , u e , y e : � � � ˙ x e = f x e , u e = 0 , � � y e = g x e , u e . Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization around equilibrium point Nonlinear systems have (several) equilibrium points x e , u e , y e : � � � ˙ x e = f x e , u e = 0 , � � y e = g x e , u e . Linearizing in the region of ( x e , u e , y e ): x = x e + ∆ x , u = u e + ∆ u , y = y e + ∆ y , Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization around equilibrium point Nonlinear systems have (several) equilibrium points x e , u e , y e : � � � ˙ x e = f x e , u e = 0 , � � y e = g x e , u e . Linearizing in the region of ( x e , u e , y e ): x = x e + ∆ x , u = u e + ∆ u , y = y e + ∆ y , with ∆ x , ∆ u and ∆ y sufficiently small. Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization around equilibrium point Nonlinear systems have (several) equilibrium points x e , u e , y e : � � � ˙ x e = f x e , u e = 0 , � � y e = g x e , u e . Linearizing in the region of ( x e , u e , y e ): x = x e + ∆ x , u = u e + ∆ u , y = y e + ∆ y , with ∆ x , ∆ u and ∆ y sufficiently small. Linearizing is done via first order Taylor expansions: � dx dt = d ∆ x = f ( x , u ) = f ( x e + ∆ x , u e + ∆ u ) , dt y e + ∆ y = g ( x , u ) = g ( x e + ∆ x , u e + ∆ u ) . Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Example: decalcification plant Used to reduce concentration of calcium hydroxide in water: chemical reaction: Ca ( OH ) 2 + CO 2 → CaCO 3 + H 2 O reaction speed: r = c [ Ca ( OH ) 2 ][ CO 2 ] rate of change of concentration: d [ Ca ( OH ) 2 ] = k V − r V , dt d [ CO 2 ] = u V − r V , dt with inflow rates k and u in mol/s and tank volume V in L. input u : inflow of CO 2 , output: [ Ca ( OH ) 2 ] Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear model and equilibrium point Nonlinear model for the given reactor: Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear model and equilibrium point Nonlinear model for the given reactor: d [ Ca ( OH ) 2 ] = k V − c V [ Ca ( OH ) 2 ][ CO 2 ] , dt d [ CO 2 ] = u V − c V [ Ca ( OH ) 2 ][ CO 2 ] , dt y = [ Ca ( OH ) 2 ] , with two state variables: x 1 = [ Ca ( OH ) 2 ] and x 2 = [ CO 2 ]. Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear model and equilibrium point Nonlinear model for the given reactor: d [ Ca ( OH ) 2 ] = k V − c V [ Ca ( OH ) 2 ][ CO 2 ] , dt d [ CO 2 ] = u V − c V [ Ca ( OH ) 2 ][ CO 2 ] , dt y = [ Ca ( OH ) 2 ] , with two state variables: x 1 = [ Ca ( OH ) 2 ] and x 2 = [ CO 2 ]. The equilibrium point ( k eq , u eq , x 1 , eq , x 2 , eq , y eq ) of this system is: Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Nonlinear model and equilibrium point Nonlinear model for the given reactor: d [ Ca ( OH ) 2 ] = k V − c V [ Ca ( OH ) 2 ][ CO 2 ] , dt d [ CO 2 ] = u V − c V [ Ca ( OH ) 2 ][ CO 2 ] , dt y = [ Ca ( OH ) 2 ] , with two state variables: x 1 = [ Ca ( OH ) 2 ] and x 2 = [ CO 2 ]. The equilibrium point ( k eq , u eq , x 1 , eq , x 2 , eq , y eq ) of this system is: k eq V − c V [ Ca ( OH ) 2 ] eq [ CO 2 ] eq = 0 , u eq V − c V [ Ca ( OH ) 2 ] eq [ CO 2 ] eq = 0 . Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization of the decalcification plant For small deviations near the equilibrium: d ∆ x 1 = − c V [ CO 2 ] eq ∆ x 1 − c V [ Ca ( OH ) 2 ] eq ∆ x 2 , dt d ∆ x 2 V [ Ca ( OH ) 2 ] eq ∆ x 2 + ∆ u = − c V [ CO 2 ] eq ∆ x 1 − c V , dt ∆ y = ∆ x 1 . Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization of the decalcification plant For small deviations near the equilibrium: d ∆ x 1 = − c V [ CO 2 ] eq ∆ x 1 − c V [ Ca ( OH ) 2 ] eq ∆ x 2 , dt d ∆ x 2 V [ Ca ( OH ) 2 ] eq ∆ x 2 + ∆ u = − c V [ CO 2 ] eq ∆ x 1 − c V , dt ∆ y = ∆ x 1 . � ˙ x ( t ) = Ax ( t ) + Bu ( t ) The resulting linear state-space model is y ( t ) = Cx ( t ) + Du ( t ): Marc Claesen System Modeling, part 2

Nonlinear systems & linearization System identification (cont) Linearization of the decalcification plant For small deviations near the equilibrium: d ∆ x 1 = − c V [ CO 2 ] eq ∆ x 1 − c V [ Ca ( OH ) 2 ] eq ∆ x 2 , dt d ∆ x 2 V [ Ca ( OH ) 2 ] eq ∆ x 2 + ∆ u = − c V [ CO 2 ] eq ∆ x 1 − c V , dt ∆ y = ∆ x 1 . � ˙ x ( t ) = Ax ( t ) + Bu ( t ) The resulting linear state-space model is y ( t ) = Cx ( t ) + Du ( t ): � 0 � c � � d [ Ca ( OH ) 2 ] c � � [ Ca ( OH ) 2 ] � � V [ CO 2 ] eq V [ Ca ( OH ) 2 ] eq dt = − + u ( t ) d [ CO 2 ] c c 1 V [ CO 2 ] eq V [ Ca ( OH ) 2 ] eq [ CO 2 ] V dt y ( t ) = [ Ca ( OH ) 2 ] Marc Claesen System Modeling, part 2